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1、ALTERNATIVE MODES FOR TEACHING SCHOOL MATHEMATICS: A SYNOPSIS_HOWARD A. PEELLEUniversity of Massachusetts, Amherst, MA 01003hapeelleeduc.umass.eduA variety of modes are profferred as alternatives for teaching mathematics in schools. Each mode is described briefly, along with general purposes, advant
2、ages and disadvantages. Combinations of modes are suggested, general issues identified, recommendations offered, and feedback from teachers summarized.IntroductionMathematics teacher educators can provide prospective and in-service teachers with a repertoire of teaching modes to help them develop st
3、udents mathematical abilities. (Here, the term mode refers to a way of structuring students learning environment for teaching purposes and is used to distinguish modes of teaching from methods of doing mathematics and strategies for problem solving.) While the National Council of Teachers of Mathema
4、tics principles and standards acknowledge that there is no one right way to teach, its vision does not specify alternatives to traditional instruction other than teaching students alone or in groups 1. This paper outlines various modes for teaching mathematics, listed in increasing order of student
5、group size and roughly from student-centered to teacher-centered: EXPLORATIONPAIRED COACHING INDIVIDUAL THINK ALOUDBRAINSTORMINGPROBLEM POSINGINTERVIEWFAMILYINCUBATIONGAMINGLARGE GROUPCOMPUTERSMALL GROUPSPRESENTATION General PurposesCommon to all these modes are a dozen general purposes, seen from t
6、he teachers perspective and linked to NCTMs principles and process standards 1:(1) Practical Purpose: To teach mathematics in a suitable setting within constraints of time, space, and resources. T Note: All modes are intended primarily for use in the classroom unless indicated otherwise.(2) Technolo
7、gical Purpose: To use appropriate technology for enhancing teaching and learning. K(3) Pedagogical Purpose: To engage students in actively studying the curriculum. T,C(4) Problem-Solving Purpose: To motivate students to apply their skills and knowledge to solving mathematical problems. PS(5) Cogniti
8、ve Purposes: To stimulate students to think about mathematics; to instill careful reasoning; and to help them understand relevant concepts and methods. RP(6) Affective Purposes: To build students positive attitude toward mathematics; to reduce math anxiety; and to nurture the joy of problem solving.
9、 (7) Interactive Purpose: To encourage students to communicate in mathematical terms.CM(8) Learning Purposes: To engage students in active learning; and to develop their skills and knowledge. L(9) Metacognitive Purpose: To develop students ability to monitor, control, and reflect on their cognitive
10、processes. L,A(10) Cultural Purposes: To respect students individual differences, heritage, values, and beliefs about mathematics; to include social, economic, and historical perspectives; and to promote equity in math education. CN,E (11) Assessment Purposes: To record students efforts; to assess t
11、heir progress; and to uphold school, state, and national standards. RN,A Note: All modes require individual student reports.(12) Real-world Purposes: To develop students appreciation for life skills involved in mathematics education; and to acknowledge relevant application areas and career opportuni
12、ties. CNNCTM Principles:NCTM Process Standards:E = EquityPS = Problem SolvingC = CurriculumRP = Reasoning and ProofT = TeachingCM = CommunicationL = LearningCN = ConnectionsA = AssessmentRN = RepresentationK = TechnologyModesIn order to help teachers choose a particular mode, each mode is described
13、below in a synopsis, along with salient advantages and disadvantages (using the same numbering as for the purposes above). Full descriptions of all modes, including recommended grade/level, time frame, special purposes and detailed operational guidelines for students and teacher, are given in 2.EXPL
14、ORATION MODESynopsis: Each student selects a mathematical topic or problem or puzzle to explore, discovers as much as possible about it, and prepares a map of what s/he finds. (This map is a guide to its important features.) Students share their maps with each other and then with the teacher, who co
15、nfirms what actually needs to be learned.Note: This may also be called Investigation or Discovery mode and can be structured with specific activities for students to follow in stages.Advantages and Disadvantages: (1) Students may work in their own chosen space - library, computer lab, or home. Yet i
16、ts easy to lose focus and to lose track of time. (2) Students can seek related information using browsers and search engines via Internet, a virtually unlimited resource. Yet they may become distracted by extracurricular material. (3) This mode is good as a warm-up homework assignment; good for intr
17、oducing a new topic informally; and good for hands-on activity. Yet without teacher control, students can fool around and might need content scaffolding. (4) It allows field work involving real data and is an open-ended opportunity to investigate, experiment, and play (without having to solve given
18、problems). Yet some students flounder due to lack of structure; some drown in too many possible causes and effects. (5) Students can start thinking naturally and build their own cognitive structures gradually. Yet its hard to overcome preconceptions, misconceptions, and mental blocks alone. (6) Expl
19、oration is emotionally comfortable; there is no overt pressure; nobody is watching or demanding immediate results. Yet some students become frustrated when they cant make progress and may give up easily. (7) Students can exercise their inner voices, debating internally about what to do next. Yet the
20、 debate may cause indecision. (8) Curiosity is nourished by discovery; there are potentially endless challenges. Yet more questions than answers may arise; and there is no guarantee that students will learn the underlying mathematics. (9) Students can build a sense of ownership for their findings. Y
21、et their metacognitive skills may not be developed enough to guide them well. (10) Students can share cultural aspects of mathematics by drawing a map for others to use. Yet they may just represent their own perspective. (11) Students can take pride in self accomplishment. Yet the teacher cant asses
22、s them very well without direct observation. (12) Life skills include: investigation, experimentation, heuristic reasoning, and independent decision-making. Yet life isnt just a bowl of exploration!INDIVIDUAL MODESynopsis: Each student works alone to study a topic or solve a math problem and annotat
23、es his/her own work. The teacher provides a list of the skills and knowledge involved to identify which were actually used and which need improvement.Advantages and Disadvantages: (1) This mode presumes a quiet and convenient place to work. Yet there is limited private space in the classroom; and in
24、trusions are inevitable at students homes. (2) It is suitable for use of calculators or personal computers. Yet some students cant afford them. (3) Individual mode is commonly used for homework, drill and testing. Yet it is an overused mode with no teachable moments for the teacher. (4) The student
25、has control, can work at his/her own pace, and can focus on the problem. Yet, without help, it is often hard to start, hard to get unstuck, and easy to give up. (5) Mental discipline is required; and writing annotations may improve understanding. Yet students often just rush to get an answer; there
26、is only one source of ideas, no check against wrong thinking, and no teacher to undo misconceptions. (6) Some students are more comfortable working individually; success boosts confidence. Yet lone failures can damage self-esteem. (7) Students can tap their inner voices; and annotations provide a go
27、od basis for communication. Yet there is no real interaction, no verbalization, and no one to urge them on; so many students prefer collaboration. (8) Students can find out privately (without embarrassment) what they dont know or cant do. Yet they can become discouraged if there is too much to be le
28、arned. (9) By annotating their own work, students must reflect on it; students can develop a sense of ownership - especially if they are successful. Yet it is human nature not to reflect on failure. (10) Students tend to work in their own established sub-cultural context of family and friends. So th
29、ere is little incentive to consider a larger cultural perspective. (11) The teacher can diagnose what skills and knowledge students need from their annotations and reports. Yet students may believe that its not their job to assess their own work. (12) Life skills include: test-taking, independent th
30、inking, organization, time/energy management, self-discipline, responsibility, and perserverance. Yet many people in the real world would rather not work alone.PROBLEM POSING MODESynopsis: Each student is invited to propose new problems and to share them with other students.The underlying mathematic
31、s must be described (separately), and problem presentation must be considered, that is, how to present a problem - its context, wording, illustrations, etc. - for a particular audience.Note: A new problem here means new to the student, not necessarily an original problem. Advantages and Disadvantage
32、s: (1) Manipulative materials may be suitable for this mode (depending on type of problem). Yet it is a chore to store and retrieve them. (2) “Inspiration” software might help students get started and connect initial ideas. But such technology is not commonly available in schools (yet). (3) This is
33、a very student-centered mode; the teacher is free to observe or participate; its particularly good for reinforcing problem-solving skills and knowledge - perhaps on a Friday in review for an impending test; and its a good opportunity for creative students to shine. Yet most students are not used to
34、it and have difficulty getting started; its also hard to detect if students are on task. (4) Students may produce some interesting problems. Yet many student-posed problems are not very mathematical or not relevant or too silly or too hard or just canned imitations. (5) Posing problems involves both
35、 creative and systematic thinking; it can spark insight and solidify understanding. Yet some students shut down mentally because it seems too challenging. (6) There is no immediate pressure to solve problems, which allows students confidence in their mathematical creativity to grow. Yet some student
36、s worry that their posed problem isnt good enough. (7) Communication skills are involved in writing and editing problem statements, as well as in explaining a problem to another student. Yet some students are reluctant to share their posed problems. (8) This mode can motivate students to review rela
37、ted mathematical skills and knowledge. Yet even motivated students may find it hard to develop specific problem-posing skills. (9) Since students clearly own the problems, they can realize that other problems have ownership too. Yet they may have difficulty judging how hard a problem is; and may ina
38、dvertently reinvent problems. (10) This is a good opportunity for students to see what problems others create as well as to express their own cultural identity in a problem statement. Yet they may be conditioned to imitate what they have seen in textbooks. (11) The teacher can select appropriate pro
39、blems for tests based on collected student-posed problems. Yet they may not represent what the students actually know. (12) Life skills include: inventing, designing, and teaching. Yet teachers rarely pose problems themselves; indeed, there are not many opportunities to do actual problem posing in t
40、he real world.INCUBATION MODESynopsis: Students consider a problem or work on a project over an extended time period. They work off and on, whenever their interest arises or after their ideas have developed. The teacher may request progress reports.Advantages and Disadvantages: (1) This mode is easy
41、 to accommodate because it moves problem solving out of the classroom and onto a “back burner”. Yet its a big change of pace from typical (next-class) deadlines. (2) Students can use resources from the Internet and personal calculators/computers whenever and wherever available. Yet some students don
42、t have convenient access to such technology. (3) Incubation is well-suited for an untimed test or large projects which require a lot of time; and the teacher can show trust in students. Yet undisciplined students may not get mobilized in time. (4) Students can work in surges, get to know the problem
43、/project well, and seek multiple solutions. Yet procrastination is common; and there is no guarantee the problem will be solved. (5) Slow pace allows careful, rigorous thinking; errors die out; students can form mental connections; and subconscious creativity can blossom. Yet real-world distractions
44、 may cause students to forget the problem/project. (6) With no pressure to work right away, students can relax and release their negative emotions. Yet it is frustrating for those who feel they cant do it and arent making progress. (7) Students may interact with others if they wish. However, if they
45、 dont, they wont get help or feedback. (8) Students are given ample time to develop understanding naturally (like a seed germinating). Yet students may not be self-motivated enough to dig in and learn the necessary content. (9) There is plenty of time for reflection here. Yet students are conditione
46、d to get the answer and reach closure; misconceptions deepen with time; and the problem/project can become haunting. (10) Students can appreciate how valuable other perspectives are - especially when they are stuck for a long time. Yet they may not seek extra cultural information outside of class on
47、 their own. (11) After Incubation mode concludes, the teacher can determine which students really cant do the work and, consequently, confirm what needs to be taught or reinforced. Unfortunately, this mode doesnt benefit quick problem solvers. (12) Life skills include: patience, perserverance, respo
48、nsibility, and time management. Yet the real world often imposes short deadlines.COMPUTER MODESynopsis: Students use (micro)computers or calculators directly to solve a problem or construct a programming project. This may involve software tools such as: programming languages (e.g., APL, BASIC, C, J, Logo, Pascal) ; mathematical software packages (e.g., Mathematica, Maple, Derive, MATLAB, Mathcad); spreadsheets (e.g., Excel); graphics utilities; simulation and modelling applets; co
