# 1988 – 30(1)

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1988 – 30(1)

ECTOR Journal of the British Columbia Association of Mathematics Teachers Volume 30 Number 1 Fall 1988 IN& I I 11111 New from Nelson Canada INTRO 11 MATHEMATICS John Kiassen West Vancouver Secondary School Intro 11 Mathematics, written expressly to meet the needs of British Columbia Students taking Introductory Mathematics 11, will focus on the Intended Learning Outcomes of the curriculum. Successful completion of this course will enable students to be eligible for Mathematics 11. Table of Contents MATHEMATICS 1. Rational Numbers 2. Radicals 3. Polynomials 4. Equations. 5. Geometry and Reasoning 6. Analytic Geometry 7. Linear Systems of Equations 8. Factoring 9. Rational Expression 10. Quadratic Equations 11. Trigonometry Intro II Mathematics Student Text 17.602341.0 uuI • Spec ____ F or The For more information, please contact your local Nelson Sales Representative. Janice Bobroske or Peter Donkers 11720 Voyageur Way, Unit #5, Richmond, B.C. V6X 3G9 (604) 273-6856 B.C. Association of Mathematics Teachers 1988-89 Executive Committee President and Newsletter Editor Ian deGroot 3852 Calder Avenue North Vancouver, BC V7N 3S3 H: 980-6877 S: 985-5301 Journal Editors Tom O'Shea 249 North Sea Avenue Burnaby, BC V5B 1K6 H: 294-0986 0: 291-4453/291-3395 Vice-President Ivan Johnson 1955 Larch Street Vancouver, BC V6K 3P2 H: 733-2168 0: 437-4511 Stewart Lynch 2753 St. Georges Avenue North Vancouver, BC V7N 1T8 H: 984-7206 0: 985-3181 Secretary Adrienne Hill T4008 - 3980 Carrigan Court Burnaby, BC V3N 4S6 H: 421-4682 0: 463-4175 Treasurer Grace Fraser 2210 Dauphin Place Burnaby, BC V5B 4G9 H: 299-9680 0: 590-2255 Membership Co-ordinator Richard Longman RR 4, 5135 Chute Lake Road Kelowna, BC V1Y 7R3 H: 764-7856 Past President & BCCUPM Representative John Klassen 4573 Woodgreen Court West Vancouver, BC V7S 2V8 H: 926-8005 S: 985-5301 NCTM Co-ordinator Tom Schroeder 3626 West 26th Avenue Vancouver, BC V6S 1P1 H: 224-8713 0: 228-5319 Primary Representative Susan Higginbottom 2653 Harrier Drive Coquitlam, BC V3E 2A5 H: 464-6549 0:594-2474 Intermediate Representative Mignonne Wood 7736 Broadway Street Burnaby, BC V5A 3X6 H: 421-5836 0: 420-3619 Member-at-Large & 1989 Summer Conference Chair Dave Ellis 2086 Newport Avenue Vancouver, BC V5P 2H8 H: 327-7734 0: 261-9364 THE B.C. ASSOCIATION OF MATHEMATICS TEACHERS PUBLISHES VECTOR Membership may be obtained by writing to the B. C. Teachers' Federation 2235 Burrard Street Vancouver, BC V6J 3H9 Rates for 1988-89 are as follows: BCTF........................................................................$25 BCTF associate......................................$25 + $ 40 BCTF membership fee Student (full-time, university students only) ......................................$10 Fee for publications (available to non-BCTF members) .......... .....................$37 Notice to Contributors We invite contributions to Vector from all members of the mathematics education community in British Columbia. We will give priority to suitable materials written by B.C. authors. In some instances, we may publish articles written by persons outside the province if the material is. of particular interest in British Columbia. Contributions may take the form of letters, articles, book reviews; opinions, teaching activities, and research' reports. We prefer material to be . typewritten and doublespaced, with wide margins. Diagrams should be camera-ready. We would appreciate a black-and-white photograph of each author. If feasible, the photo should show the author in a situation related to the content of the article. Authors should also include a short statement indicating their educational position and the name and location of the institution in which they are employed. 2 Notice to Advertisers Vector, the official journal of the British. Columbia Association of Mathematics Teachers, is published four. times a year: fall, winter, spring, and summer. Circulation is approximately 600, mainly in B.C., but it includes mathematics educators across Canada. Vector will accept advertising in a number of different formats. Pre-folded 21.5 x 28 cm promotional material may be included as inserts at the time of mailing. Advertising printed in Vector may , be of various sizes, and all must be camera-ready. Usable page size is 14 x 20 cm: Rates per issue are as follows: Insert: $150 Full page: $150 Half page: ., $ 80 Quarter page: $ 40 Deadline for submitting advertising for the spring and summer issues are February 24, 1989, and April 28, 1989, respectively. Inside This Issue 4 5 From the Editors ................................................Tom O'Shea Letters .................................................................... Mathematics Teaching 6 Active Learning in Mathematics ................................John V. Trivett 14 Word Problems + Calculators = Motivation + Success........Katherine Willson 16 The Adams River Sockeye Run of 1986 .....................David R. Parkinson 18 Students An Incredible Resource (2)..............................Zoe Wakelin 20 MathTalk—An Electronic Conference ............................Stewart Lynch Mathematics Issues 23 The NCTM Standards Document Mathematics and Computers 34 Functions as Models of Reality .................................Brian Copeland 45 Secondary School Calculus .................................Harold Brochmann Miscellaneous 54 Report on the 27th Northwest Mathematics Conference "Mathematics for the '90s .......................................... Ian deGroot 56 The SFU Master's Program in Secondary School Mathematics Education ..........................................Tom O'Shea From the Editors Tom O'Shea ell, the 27th Northwest Conference W has come and gone. For me, it was one of the best we have hosted in British Columbia. Facilities were good, speakers were excellent, and the number of people attending exceeded most expectations. Congratulations to the organizing committee for a fine job. Again, we encourage presenters to put their ideas on paper in the form of articles for Vector. Our apologies to Jim Vance, from the University of Victoria. We inadvertently left his name off the article in the last issue of Vector describing the Northwest Conference. In this issue, John Trivett begins by considering what it means to be active in learning mathematics. John has been interested in this area for many years, and, as usual, his perceptions are insightful and well worth considering, especially as we implement the new activity-oriented curriculum. Katherine Willson shares her experience in asking Grade 1 children to create a problemsolving class booklet. It's a nice example of how the calculator can become a natural part of the mathematics classroom. David Parkinson follows and shows how mathematical procedures can be found in problems related to the local environment. Zoe Wakelin shows how having students create their own problems in geometry can reveal • lot about their thinking. A side benefit is • collection of problems that the teacher can use later for worksheets and quizzes. Stewart Lynch steps out of his role as coeditor and explains the new electronic conference for mathematics teachers called MathTalk. We're both on the system, and we welcome suggestions for future editions of Vector. We hope to publish excerpts from the conference to show the benefits of participating. The NCTM Standards document is one of the most important developments in mathematics education in North America. It will probably mark a turning point for the U.S.A. in developing a national consensus on what mathematics should be taught in U.S. public schools. Some of us have had a chance to respond to an initial draft, but many math teachers are unaware of what it is and what it means. We are reprinting a portion of it here to give it the widest possible circulation in B.C. In the mathematics and computers section, Brian Copeland continues a series of articles on using the computer to teach mathematics. Here he examines how the computer can be used to help understand the behavior of functions that are particularly useful in modelling natural phenomena. Harold Brochmann begins a series of three articles on teaching calculus in secondary schools. In this issue, he argues persuasively that the first step should be to show what calculus can do, rather than present .- formulas to show how to do it. In the final section, Ian deGroot reports his perceptions of the 27th Northwest Mathe- matics Conference. Finally, I take a great deal of pleasure in announcing that Simon Fraser University has developed a Master's Program specifically designed for secondary mathematics teachers. We plan to begin the program in September 1989, so if you are thinking about graduate work, now is the time to get started. I welcome all enquiries by mail, telephone, or electronic mail. -r The Editor: I read "Calculus in the Secondary School" in the Spring '88 edition of Vector with great interest. Thank you for an excellent treatment of .a complex topic. I would like to add to the debate a dimension that is usually overlooked but is relevant to B.C. teachers today. Teachers care about the success of their students. Many teachers say to themselves, "All the other schools areteaching calculus. I know because that is what the students who were in my class last year tell me. Therefore, if I want my students to have a fair chance, we had better introduce a calculus course in our school." It is easy to explain the source of this perception, and we could even show that it is not valid. At least we could a year or two ago.. It went something like this: "All the students who ask questi9ns in the calculus class seem to have had some background in it. But only those who know enough to ask questions speak out. The instructor responds to those who ask and assumes that the others know more than that. Then my ex-student gets lost and is convinced that it is because I didn't teach him any calculus in high school." I used to argue that the problem was not real and the simple solution was to keep reminding the universities that calculus was not part of the secondary school curriculum. Now it seems that the reality is catching up to commonly held perception. Maybe it is too late to argue whether calculus is 'better taught here or there. We should probably agree on a course, teach it well, and hope the postsecondary schools adjust their courses appropriately. Sincerely, Jake Penner MATHEMATICS TEACHING Active Learning in Mathematics John V. Trivett John Trivett is a professor emeritus in SFU's Faculty of Education. ere is an introduction to some of the H important aspects of active learning in mathematics. They apply to all students at any grade, whether they be deemed by some to be slow, or fast, dull or bright. The examples quoted can be multiplied almost without limit, and they have to be for adequate classroom implementation. Discovery In a sense every student has to discover everything. The teachers' mere explanations, looks, or activities do not necessarily get a reaction by a student, let alone a hoped-for reception. The text may be handled and so may physical aids, but there have to be continual acts of volition by every student to trigger some kind of reception in the student's eyes or mind. Varying levels of reception are needed for understanding, however. That a fraction is conventionally written in the form -- needs little discovery. It is seen and almost immediately accepted by the youngest of students in school. That it represents not only a particular element but the whole of a particular class of ordered pairs of whole 6 numbers needs more and more discovery with a different quality and quantity of what has to go on within the person to make it as secure as the symbol seen with the eyes. In mathematics teaching, some have traditionally acted as though simple surface-level discovery is all that is needed. We explain, we do examples, we show how, and we get the children to practise on paper. We try to define subtleties, and when we see the pupils failing, we tend to conclude that it is because some are not capable. We call them inattentive, lazy, slow, and by using more simple words, we suggest that it is they who are at fault. Seldom do we think that we teachers have not yet attained the highest forms of communication and activities to induce what is needed to have all students learn the mathematics. If, therefore, discovery is a fact in learning processes, not just the name of a method, we surely have to enquire more into the ways and means by which students are compelled to take responsibility for their-own learning, for no one ever did anyone else's. Teachers must minimize correction from an apparent authoritative standpoint and insist on being an authority only where the student cannot be. That can be condensed to just two functions: orchestrating the activities of the group and telling the children what are the agreed conventions of symbols and form. In all other matters, in the widening consciousness of the fundamental facts, the pupils must be helped to discover at first hand, and although little attention has been paid to this under the weight of verbal and written tables and formulae to be learnt, every man and woman knows the effect 'of a general understanding into which many details fit, compared with the knowledge of many details with no connecting substratum of understanding. Teachers need to be learning facilitators, creators of learning environments, flexible ones, appropriate to the tasks, to the actual students involved and capable of minute-tominute changes. As a result of wisdom and knowledge, using the art and craft of the trade, the teacher of mathematics in particular initiates activities such that 'anyone engaged in them, can't help being affected mathethe by the concentration , noflatent matical ideas and cocepts built in. It is arranged deliberately that the chances are very high that no student can avoid the inundation and, what's more, not enjoy it!' The environment will include the use of physical aids, the writing on paper, the usual symbol work, and texts, but it will also take into consideration' the inner consciousness of every person in the room. No system can be ignored. Inner systems of humans must be recognized as essential concomitants of the learning environment: thoughts, wishes, day dreams, mind wanderings, feelings. These constitute the.stuff of existence as much as, or more than, the manipulative aids, the books, the diagrams, and the mathematics we aim for. It is the teacher's responsibility to watch the learning process and to introduce creative conflict at appropriate moments. Individuality Students have to do their own learning, with their own insights, past, and present as they see it. These are the filters through which pass all that occurs in our math lessons. We had better be aware that this is so. Each students is unique and may differ much in his or her reception of something public from all the other receptions of the in'dividuals in the group. It is impossible that two people receive the same thoughts,' the same understanding,no matter how the teacher tries. Whereas one child happily accepts a sequence, for example: log 20 = 1.3010 log 2 = 0.3010 log .2 = 1 + 0.3010 log .20 = 2 + 0.3010 another's difficulty may be that in the third and fourth lines there are + signs, but in the other lines there are not. The difficulty evaporates as soon as he or she sees that one could write the equivalents: log 20 = 1 + .3010 log 2=0+.3010 log .2 = 1 + .3010 log .02 = 2 + .3010 etc. Now there is a balance of the forms. If the teacher does not understand a possible obstacle here and cannot help, the student may well feel he or she is just incapable of understanding where apparently the others do,. Another example, from elementary school Student A. 2.48B. 2.48 —1.69 —1.69 1.21 C. 2.48 —1.69 1.21 .81 D. 2.48 —1.69 .79 Although child D is "right," it does not follow that we know what the child did, or how he or she thinks. Correct answers are frequently written for the "wrong" reasons. What is needed is a suspended judgment, not a stamp of approval or disapproval. If we can find out what each child really did (and a good way of doing this is to ask and listento replies!), this is what might emerge: A says "9 from 8, can't, so 8 from 9 is 1 ("always do what you can") 6 from 4, can't, 4 from 6 is 2 etc." B says "9 from 8, can't, so 8 from 9 is 1 16 from 24 is 8." C says 9 from 8 is 1, so 6 from 4 is 2 etc." D says "9 from 48 is 39, put down 9 and carry 3 3 from 20 is 17, but 1 from 17 is 7 Put down 7." That's the kind of tiny iceberg top of what goes on within individuals. It occurs all the time, in all lessons, with everybody. As the understanding that this is so grows, together with an increase in knowledge of more and more possibilities within this vast, commonly unexplored field, every teacher can open new vistas in his or her approach to teaching. The consequences are greater self-respect and successful learning for all students. Individualism also is a fact, not a method, not an opinion. We cannot choose to treat humans non-individually and succeed. If an "individual approach" appears to be only a fad, in which it is assumed that organization is at the bottom of individualism, and if this passes next year to some new urge that has all students reacting the same way, people will still go on acting individually. Even if they are overwhelmingly herded and made to jump through the same hoops of authority, they still won't oblige except for a short while. Humanity's eternal quest for freedom will be maintained despite temporary setbacks. Games Children and adults like playing games. Games are concentrated, challenging, enjoyable activities that can be given up at will; whereas work is more unwilling, and more pressure is accepted before it can be abandoned. Children often like to play games with physical objects, and they have shown this in their preschool experience. However, games are not restricted to the use of physical objects nor competition against others. Word games, written games, and games with oneself and with textbooks can all be rewarding. Some of the physical games are quite new in mathematics lessons. Some teachers have for years had children apply mathematical and arithmetical principles to measuring, shopping, and surveying, but nowadays they also have for use many physical materials that lead to awareness of mathematical principles by the very act of using them and becoming more conscious of what one is doing. Colored rods, attribute blocks, multibase blocks, geometry models, geoboards, pebbles and counters,. mirrors, cutout figures, films, one's own body and body movements—all provide opportunities to be taken by good teachers. numbers, addition and multiplication, and more of the familiar list of what we want kids to know. As an example, consider the use of the socalled attribute blocks, which are of different shapes and different colors. A red triangular block and a red square are alike because they are both red, but they are different in shape. When we use blocks of other shapes, other colors, and use some with different thicknesses and even different materials, then we have pieces for many games that demand in the very acts of playing understandings of differences, equivalences, intersections and union, elements, and subsets. Such notions are essential to a proper understanding of By the use of games, improved mastery of the traditional math and arithmetic can be attained. Instead of laborious, non-related facts that somehow have to be memorized, the games approach involves thoroughness and intensity, and it subtly demands every student's ongoing dedication. Games with physical objects are only one type. There are many others, so here we illustrate one kind of game that seems quite traditional: Continue this web. From any name, draw an arrow, and write another name for the same number which comes to your mind. Have a reason why you went from one name to the next. 8+6 - (8-3)+9 7+7 - 2X7 - 1/2X4X7 9+5 t (7-2)+9---(6-1)+9 - 5+9 14 - 10+4 5+3X3 15-1 / 5 X., 1 +3 X 3 16-2 / Do not ally the word games with something frivolous, a pastime, something to do "after work." The most played games are very demanding in energy, devotion, thought, and time, and they increase learning and powers. To try to meet mathematics as a series of games is not to drag it down to the wasteful, trivial filling of time. To try to meet mathematics as a series of games is to elevate certain mental activities to the level of some of humanity's finest activities: exploring space, serving others, living itself. Mathematics Mathematics is concerned with the dynamic use, in the mind, of relations and relations of relations, with some applications to social and economic life. Arithmetic is one of its branches having a main emphasis on the renaming of numbers 9 by means of algorithms, tables, figures, calculations, and other devices. To learn that 2 + 3=5 may only be a matter of remembering that "5" is a word that is an acceptable substitute for the phrase "3+2," and "6" is not the accepted word. To recognize, however, that 3+2=2+3 implies much more. Now we get a hint of a basic principle (commutativity), and this may lead to other interests. Alternatively, it gives us a pattern to get new names for numbers we have not met before. (4576+687 surely must equal 687+4576 even if the former has not been confronted previously.) To write or say 2+3=5 is a convention, a convenience. The symbols, as such, give no clue to what they stand for. It is just as easy to accept the writing 2 + 3 = 7, but historically that is not what was developed, so we think that 2+3=5 is correct and find 2+3=7 uncomfortable. The meanings behind the symbols, however, are not conventions. They are based on fact over which we have no control. We can only be ignorant of the fact. The fact is embedded in life, and every one of us is capable of firsthand experience and enlightenment. We therefore encourage mathematics partly to understand and use the ideas, the concepts, the relations, and the facts, and partly to help us manoeuvre the old arithmetic that is still vital and valuable. On the whole, we do not question the kinds of activity suggested by the use of mathematics textbooks so long as the teacher appreciates that the one printed form, the book order, and the style of presentation is not necessarily right for any student whatsoever. Those mass-produced aids show the sophisticated outcome of years of thought and centuries of argument and symbol evolution. They must be seen as such, used for the embodiment of desirable aims maybe, but approached along different and unsophisticated paths. They rarely exhibit the form of 10 progress precisely needed by any child to reach many of the same conclusions. Example Usually in textbooks, the addition of fractions is dealt with before the operation of multiplication. One student pointed out that addition is SO: -- + -- 3 7 — (2x7)+(3x5) (3X7) 29 21 and multiplication is thus: 2 x - 2X5 - 10 3 7 — 3X7 21 Addition, it seems, implies three multiplications and one addition. Multiplication needs only two multiplications. On what grounds, he asked, is addition considered easier than multiplication and taught first? Had he a point? Integration If math is indeed concerned with relations and activities with relations, then it is essential that students constantly have experience that relates things for them. It is their meaning that is vitally important. Continuous progress will be unique for each individual, and the relations that are explored will not by any means be confined to that portion of the school day called the math lesson. Relations are abstracted by all humans in all kinds of situations. The math lesson will emphasize them and show what can be done with them, and greater power in their use will be gained. But the existence of relations must be from all subjects studied, from all life. Appreciation of integration across the curriculum is therefore important. Integration is also needed within the math study itself so that growing experience is had in relational activities that will become second-nature and pervade all the educational scene. Within the math study, such integrating, creative aspects can be dealt with in many ways. Here are two, briefly: 1. Reverse the usual task of getting an answer to a problem. Instead, get a problem to fit the answer, e.g., x=7. Write some other equations of which x = 7 is the solution. E.g., The sum of four numbers is 2489. What are the four numbers? 2. Rather than set word problems, give an equation, and, from it, invent an appropriate social situation. E.g., 3y+4=19. What situation in a grocery store might fit this? Across the curriculum, apart from some obvious use of number work and equations, are rich, powerful mathematical ideas of relations, differences, sets and subsets, transformations, symbolisms, reversals and repetitions to explore and use. All occur and have their importance in social studies, art, science, and music. E.g., Grade 1 If children can read and write in some form, the words "step up pat," they can also read and write all six permütations of the words. They can be asked to do so, with the teacher expecting them to do it; They can also be expected to read every one of the six sentences and know whether they make sense or not. Grade 6 In discussing "humans," students can be helped to list equivalences and differences of humans and see intersections of attributes. This leads to an improved awareness of racial similarities and differences, all within the same human species. Humanness No mathematics lesson ever attended took place without human beings being present. Mathematics. may be devoid of feelings, opinions, and other vagaries of everyday life, but mathematics learning situations are not, for the humans present have such attributes all the time. In the learning of the subject, therefore, there are facts of this kind to be taken into full consideration by the teacher: the need for understanding, respect for others, tolerance, frustration, tiredness, hope, lack of communcation, annoyance, satisfaction. Unless teachers are aware of these as being present in every lesson, in every human, then a large part of what is really going on in the lesson will be completely missed, and the teachers will be forced to resort to the traditional rationalization of "The students won't attend," or "They are slow and incapable." These are very real parts of everyone's life, and we must certainly recognize their existence and the part they may be playing. Because of such awareness and allowance, we should help the students commune with each other and with ourselves, since we shall no longer depend on outward signs alone that give little clues to the thinking and the perceptions going on within. Perhaps the future progress of all learning depends solely on the harmonizing of what has to be passed on to the new generation within the context of the "species-specific mode" of every learner human in this case. It is fruitless to try to teach a dog to do what a dog is incapable of doing. The same holds for humans. We assume that telling is sufficient; whereas if we pause and think for a 11 moment, each of us has plenty of evidence that telling is seldom sufficient for communication for anything other than trivialities. Automatically learning by being told is not a human mode. Humans do not pay much attention to what others are saying. The mode surely suggests that we have to encourage acitivities in which it does not matter if the learners attend or not—except to themselves, which they cannot escape. E.g., if a Grade 5 pupil wants to add fractions thus: -'+ -- - 7 7 9.16 let her and encourage her to do more like it. 4 + 1216 3 1316 1+1+1+- 7 1 16 1 1 1 1316 1 1 13 (I) When she had developed other names for according to the same rules, she most probably will begin to sense a.contradiction with other thoughts she has about fractions. That contradiction within herself will provide motivation and, at least. in this case, the realization that that is not the way fractions can be combined. to act more and more asthough the learning is simple even if the organization is complex. A simple solution or a straightforward outward appearance can invariably be assumed to be misleading and wrong. There have to be discovery and non-discovery; rote learning will still play a role; there will be partial understandings and superficial insights along a spectrum. There will not be artificial discussion about group versus individual; for every group is one of individuals, and every individual is a member of many-groups. The variation of possible individual responses to every tiny situation, the variation of movements, thoughts, things to do, and the way for people to do them—all contribute to a fantastic number of alternatives that in practice, whether we like it or not, make every lesson different, no matter how hard we try to repeat or homogenize. Most of what goes on, most of the active learning, will be unseen by the members of the group. Each individual will have a concentrated view only of his/her own incoming sensations and inner thoughts. And on top of all that, there will be the mathematics, the activity with relations, numbers, space, sets, operations, functions, and the arithmetic as a by-product of some of this. The complexity is inevitably so great that it makes one wonder how we ever got the belief that we can determine in any but rough form how some piece of learning should take place... Why do teachers plan lessons rather than prepare themselves to meet, their students as they will be by the time of the next lesson? Summary What we suggest, therefore, is a systems approach, one that considers all the systems active within the learning situation, outward and inward. Much cannot be avoided, some of it can be avoided, and some can be initiated by . the teacher. Every learning, situation is a complex set of circumstances; whereas the history of education has tended 12 A balance is needed, not one that attempts to remain still but one that is nevertheless stable. Sometimes the of activity will seem to be of.. one kind but, because of the teacher's innate stability, and his or' her understanding, purposes, and leadership, there will be some vaguely perceived fulcrum about which the kinds of activity will oscillate. The mathematics classroom will sometimes appear to be a hive of physical activity. Children will be measuring, weighing, or using blocks and ropes to discuss sets. Others may be constructing tetrahedrons and investigating their rotations. Computers may be in use, or some pupils may be outside the school with angle meters doing elementary surveying. Sets of cubes, colored or plain, may be seen, although the students using them may really be discovering the relations inherent in successive cube numbers. All the fun of the fair may seem the order of that day. On other days, or with some children perhaps during the same time period, the activity will be with paper and pencil, and the appearance will be traditional. The facts concerning the consciousness of the different systems extant must lead to variety, for each learner, for the whole class, and for the teacher too. Underlying all, however, will be the commonality of an increasing mathematical consciousness for each child within a human community. This will result from the effects of an environment purposely engineered by the knowledgeable teacher in which the learning takes place intensely, individually, integratively, and joyfully. That is the "active learning in mathematics" to aim for. 13 Word Problems + Calculators = Motivation + Success Katherine Willson Katherine Willson is an assistant professor in UBC's Department of Mathematics and Science Education. aving children write their own word H problems is a highly successful strategy for problem solving. The process engages pupils in a teaching method that éombines reading, critical thinking, and the collection and organization of data (Fennell & Ammon, 1985). Several years ago, toward the end of the year, I informed my Grade 1 class that we were going to create a problem-solving class booklet, and each pupil would have a chance to be an author. The children were highly motivated, and we discussed the emphasis on solving real problems found in the classroom, school, or home. Many teachers and researchers are aware that students are highly motivated when asked to solve problems that are real to them, as opposed to contrived problems found in the textbooks. Trading stickers and hockey cards were the current rage, and many children were enthusiastic about writing problems that dealt with trading. The pupils understood the meaning of addition and subtraction and had solved many word problems throughout the year. They had the ability to read and write numbers to 100 and had learned basic facts to ten. They had never created their own problems, so I was unprepared for the results. Many children chose numbers that would result in calculations beyond their computational abilities. One child wrote the following problem: 14 One day I had 44 stickers. My dad gave me 18. How many do I have now? Although the student coukl read the problem correctly and identify it as requiring addition, he could not easily solve the problem by adding because he had not learned the addition algorithm. One solution would have been to have him act out the problem by using real stickers. However, many other students had used even larger numbers, which would have made the approach time consuming and cumbersome. One particularly bright student had written a problem to determine the number of children in the school. She knew she had to add eight twodigit numbers, but was frustrated when she could not solve the problem. When she finally solved the problem, she was able to read the total even though it went beyond the Grade 1 curriculum. Knowing that children are intrigued by large numbers and that often writing problems dealing with numbers ten or lower creates artificial situations, I should have predicted the outcome. No child in today's society Would likely possess only ten or fewer stickers. To ask that pupils limit their problems to working with numbers ten or lower is not only unrealistic, but also unmotivational. How was the dilemma solved? We had used calculators throughout the year to explore counting and place-value concepts as well as basic facts, so several pupils suggested that we use them for problem solving. After a quick review of adding and subtracting with a calculator, pupils had a choice. They could use counters (or the actual objects present in the problem) or a calculator. All children but one chose the calculator. Another restriction was added. The numbers in the problem could not total more than one hundred (in order to ensure consistency with the curriculum). •Estimation was emphasized, and several lessons on rounding to the nearest ten were taught. Answers were checked for reasonableness. ability to read and write numbers to 100 but were limited in their computational ability to working with numbers to ten. With the aid of a calculator, pupils could work with numbers to 100 and were no longer restricted by computation. They were motivated to write and solve more problems than would have been possible without a calculator. Since problem solving success is directly related to the number of problmes solved correctly, the calculator is a significant asset in the primary classroom (Suydam, 1987). What was the final result? Once children had access to calculators, their motivation and productivity increased. All wrote more than five problems, and one created twenty-three. Solutions were written on the back of the problem sheet and checked by a classmate. Favorite problems were included in our class booklet, which was frequently signed out to be taken home and shared with parents. Students were also enthusiastic about solving their classmates' problems, and they solved a large number of problems. References What were the advantages of using a calculator? Every Grade 1 pupil could add and subtract when given a calculator. The focus was on the problem-solving process, not on the computation. Students had the Bruni, J. (1982). Problem Solving for the Primary Grades. Arithmetic Teacher, 29 (6), 10-15. Duea, J., Immerzeel, G., Ockenga, E., & Tarr, J. (1980). Problem Solving Using the Calculator, In Problem Solving in School Mathematics, 1980 Yearbook of the National Council of Teachers of Mathematics. Reston, VA: NCTM. Fennel!, F., & Ammon, R. (1985). Writing Techniques for Problem Solvers. Arithmetic Teacher, 32 (1), 24-25. Suydam, M. (1987). Indications from Research on Problem Solving. In F.R. Curcio (Ed.) Teaching and Learning—A Problem Solving Focus, Reston, VA: NCTM. 15 The Adams River Sockeye Run of 1986 David R. Parkinson Dave Parkinson is a mathematics teacher in Shuswap School District. ere is a real problem related to the H Adams River sockeye run of 1986. It outlines a capture-recapture method, and all information and numbers were obtained from a fisheries officer in September 1986. I think the problem would be useful for Grades 4 to 8 mathematics classes. Background to the problem The number of sockeye that actually spawn in the Adams River is of concern to federal fisheries officers because they need to predict how many salmon will be returning four years from now. The size of this year's run was expected to be large because of the number of salmon that spawned in 1982. The number of salmon actually starting this year's run at the mouth of the Fraser River has been estimated at between 16 and 17 million. The estimate was made by counting the number of fish caught in one hour by commerical fishermen near the Fraser's mouth. That number is then multiplied by the number of hours that the fish are passing that point. Unfortunately, it is illegal to net fish for food at the mouth of the Adams. To estimate the number of sockeye out of that 16-17 million that will actually spawn, fisheries officers use a capture-tag-recover system. A seine net encircles a school of between 1200 and 1500 salmon near the mouth of the Adams, and these are tagged with a fin marker and released unharmed. This tagging is repeated until between 25,000 and 30,000 salmon are tagged. Each day between early October and early November, officers walk the river 16 collecting dead salmon and throwing them into a pile. This procedure is called dead pitching and makes sure that no salmon are counted twice. The problem 1. How could you use this information to estimate the number of salmon actually spawning in the Adams River? 2. In what year will the next big run of sockeye return? 3. Estimate the population of untagged Adams River sockeye actually spawning if 28,132 fish are netted, tagged, and released, 55,141 are dead pitched, and, of these, 556 are found to have tags on them? 4a. Use a calculator to find the "exact" number of salmon that spawned. Is it possible that this is not really the exact answer? 4b. Explain your answer to 4a. 5c. To what place value did you express the answer to 5b7 Is this reasonable, or should you round it off more or make it more accurate? Explain your answer. 5a. What fraction of the fish that entered the Fraser actually spawned? 5b. What percent of the fish that entered the Fraser actually spawned? 6. The number of cedar trees in a mixed cedar-fir forest can be estimated using another sampling procedure. Describe how the number of trees in a 50,000 hectare forest can be estimated. Also describe how the percentage of cedar trees in the stand can be estimated. BCAMT-Sponsored Conferences in 1989 MATH TECH 89 February 17, 18 Burnaby SUMMER CONFERENCE August 29, 30 Vancouver 28th NORTHWEST October 12, 13, 14 Seattle 17 Students— An Incredible Resource (2) Zoe Wakelin Zoe Wake/in is a mathematics teacher at Vernon Secondary School, in Vernon. n a geometry quiz, given to two O heterogeneously grouped classes of Math 8 students, I included the following question, which requires the upper-level thinking skill of synthesis, together with comprehension of the geometric concepts. Below are some of the student solutions I later used on worksheets and quizzes. Create a question that involves alternate interior angles, supplementary angles and an isosceles triangle to find the measure of a particular angle in the diagram. (3 marks)Find 45 Find 4 1 (Chris Lang) (Clint Ingham) Find 42 (Chris Godard) 6 Find 4 3 18 (Robyn Winskowski) Find 4 6 (Melanie Budgen) Find 4 1 0 marks No attempt at a solution. No property used appropriately. 1 mark Appropriate use of one property. 2 marks Appropriate use of two properties. Three properties used, but not all essential to the solution. 3 marks All three properties used, and all essential to the solution. (Luke Rosnau) Evaluation of this type of question is time consuming. However, working through each solution provides valuable information about the student's understanding of the geometric properties. I found the following marking scheme effective and easy to use. National Council of Teachers of Mathematics Leadership Is Vital to Improving Mathematics Education Mathematics Leaders in Secondary Schools advocates strong mathematics leadership in each school building. It is imperative that every secondary school designate a leader, frequently called the department head. This will significantly improve secondary mathematics instruction. This new position statement from the National Council of Teachers of Mathematics also stresses that these leaders must provide ongoing assistance in planning, implementing, and evaluating a comprehensive mathematics program. They must lead and assist teachers, principals, supervisors, and other administrators in the areas of curriculum, methodology and materials, student assessment, professional development, and procedural duties. Mathematics Leaders in Elementary/Middle Schools (March 1984) is a companion piece to Mathematics Leaders in Secondary Schools. This position statement also advocates that strong leadership in individual schools is essential for assuring a high-quality mathematics program. These leaders must be available as a resource person to all staff and must demonstrate knowledge, competence, and leadership in mathematics education beyond what they may be expected to teach. Mathematics leaders in all schools will have the responsibility for implementing the recommendations in Curriculum and Evaluation Standards for School Mathematics, which is scheduled for release by April 1989. Copies of both position statements are available from the NCTM Headquarters Office. 19 MathlalkAn Electronic Conference Stewart Lynch Stew Lynch is vice-principal at Argyle Secondary School, in North Vancouver. Introduction e math teachers tend to spend most W of our time alone in our classroom with our students. We seldom get the opportunity to discuss items concerning mathematics unless there is a major conference or professional-day workshop. Our discussions are limited to those within our own school or district. With the aid of a microcomputer, a modem, communications software, and the telephone line, we can now participate in a wide-ranging, intellectually stimulating, professionally rewarding exchange with dozens of. new friends and hundreds of colleagues every day. This virtual community is a group of people who may or may not meet one another face to face, and who exchange ideas through computer bulletin boards and networks. The community. members share a common interest; in our case, mathematics. This network can be used to scan and gather information on subjects that are of momentary or enduring importance, from lastminute lesson plans to long-range plans of curriculum implementation. It is a bit like a neighborhood pub or a coffee shop. We 20 can participate in hundreds of ongoing conversations, have our questions answered, and find inspiration and support. The communications network has several advantages over the old-fashioned communities of place and profession. Because we cannot see one another, we are free from prejudices about others as we read what they have to say: Race, gender, age, national origin, and physical appearance are not apparent unless a person wants to make such characteristics public. In a communications network, we can get to know people then choose to meet them. We can exchange information about one another, share and debate our mutual interests, and sometimes become friends. In this community, we go directly to the place where our particular interests are being discussed, then get acquainted with those that share our passions. In a sense, the topic is the address. You can't simply pick up the phone and ask to be connected with someone who wants to talk about the new Math 9A curriculum or about a new topic in Data Analysis; you can, however, join a computer conference on any of those topics, then open a public or private correspondence with the previously unknown people you find in that conference. You will find that your chances of making friends are magnified by orders of magnitude over the old methods of finding a peer group. MathTalk With so much activity going on in mathematics revision and implementation, the BCAMT executive thought it a good idea to have some method for mathematics educators at all levels to communicate daily. MathTalk is our forum for discussion. Xchange—The Education Network The Faculty of Education at Simon Fraser University has developed an electronic network of teachers and students throughout the province. The Faculty of Education worked with SFU's Computing Centre to design an electronic educational forum (EdForum) to serve the professional development needs of teachers in the province. Edforum carries electronic conferences dealing with issues in many subjects. Participants can also access the Groliers Electronic Encyclopedia and ERIC (Educational Resources Information Centre) and open up the exciting world of on-line information for themselves and their students. A computer conference like MathTalk is like any other conference. Computer conferences are electronic forums or meeting rooms in which participants post messages for others to read at their convenience. A variety of sessions are going on concurrently. MathTalk has several active sessions where interested participants can join in and monitor or participate. Throughout the year, Vector will provide its readers with excerpts from the various discussions taking place in the conference. If a discussion does not meet the needs of a particular individual, that participant can begin his or her own discussion. Xchange provides a resource guide to electronic mail, conferencing, and databases. The Xchange materials include a tutorial and conference directory and will be updated regularly. Xchange is also supported by a team of online moderators who will manage the electronic conferences. On behalf of the BCAMT executive, I will be moderating a conference in Ed-forum called MathTalk that is for all teachers and student teachers of mathematics who want to discuss issues concerning the mathematics curriculum. Xchange will also deliver in-service workshops, on request, to school districts and other groups. Members of an SFU in-service team conduct hands-on training sessions in electronic messaging, computer conferencing, and accessing online databases for educators wishing to extend the walls of their classrooms and enter the rapidly transforming world of electronic cothmunications. Discussions on MathTalk currently include the following: BCAMT issues Problems with implementation Primary mathematics curriculum Intermediate mathematics curriculum Geometry Data analysis Algebra Number and number operations Measurement Calculus Math 9A and 10A Technology and mathematics Resource materials The list can go on and on. MathTalk enables rapid exchange of information on issues and ideas concerning the teaching and learning of mathematics. It is designed to promote interaction among teachers of mathematics who are looking for information, ideas, support, and challenges for both themselves and their students. Teachers can exchange 21 teaching tips, lesson plans, and personal opinions of the latest books, materials, and technologies. At present, access to the SFU computer system is free. The Xchange materials, including the resource guide and regular updates, are $25 and can be obtained from Simon Fraser University by writing: What You Need To Get Started To participate in this network, you must have access to a microcomputer that is connected to a modem. A modem is a device that translates computer signals into sound waves, and vice versa, so that two computers can communicate through the telephone lines. Each computer is hooked up to its own modem. The type of computer is unimportant. Apples can talk to Commodores, Radio Shacks, IBMs, Macintoshes, and any other computers as long as both computers have modems set at the same baud rate. A 1200-baud modem is recommended and can be purchased for less than $150. Communications software is also needed to give the commands to the modem, such as what phone number to dial. Several modems come with their own software, but several integrated packages like Microsoft Works, for the IBM and Macintosh computers, have a communications module that will do the job. SFUXchange, Faculty of Education Simon Fraser University Burnaby, BC V6A 1S6 tv If you are communicating from outside the Lower Mainland, you will connect with SFU using the telephone number of the Datapac Public Dial Port nearest you. Datapac is a computer communications network that serves most of Canada and is available in most cities in British Columbia. Your call from outside the Lower Mainland will be toll free from the Datapac port to Simon Fraser University. The recent Northwest NCTM Conference was an overwhelming success. Let us keep the exchange of ideas flowing. This is a time of change in mathematics, and we need to communicate. Conferences and workshops are too few and far between. MathTalk is your link to what is going on outside your classroom. Ric it?0 I/el + 22 4XOr1% 0 MATHEMATICS ISSUES The NCTM Standards Document he NCTM Commission on Standards for School Mathematics created in 1986 was charged with developing a set of standards for mathematics in an exemplary school mathematics T program and the instructional conditions necessary for that mathematics to be learned. In addition, the commission was asked to develop standards for evaluating such a school program and evaluating of student performance in light of the curriculum standards. A draft set of standards has been prepared that reflects the output of the commission's working groups, which met several times during the summer of 1987. Membership of the working groups represents a cross-section of classroom teachers, supervisors, teacher educators, mathematicians, and researchers, all active in the work of the council. The NCTM circulated a draft of the standards and asked that mathematics educators comment on the document. The comments were collated in the summer of 1988, and a final document will be presented at the 1989 Annual Meeting of the NCTM in Orlando, Florida. The Executive of the BCAMT considers the document to be very important, and it spent a full day at its summer retreat discussing the standards and implications for mathematics education in British Columbia. We believe that the standards will have a strong impact on curriculum and teaching in the United States, and in other educational systems. Here are some of the questions we raised: How realistic are the assumptions? Are some of the beliefs just platitutdes? Which beliefs does the BCAMT Executive support? If there are assumptions that are not realistic in the B.C. scene, can we ignore them and still support the standards for a specified grade level? Even though 10 000 copies of the draft of the standards were distributed, and some reference has been made to them in other NCTM publications, we feel that BCAMT members may not yet be familiar with the recommendations. We cannot circulate the entire report, but we have chosen to reprint much of the introductory section to give the flavor of what will come to the 1989 NCTM Annual Meeting. We urge you to read this material, to consider the curriculum changes in British Columbia, and to assess whether we are in step with the NCTM, or perhaps leading. We also hope that you will obtain the final document to see what specific recommendations are being made at each of the primary, middle, and high school levels. 23 Introduction to the Standards BACKGROUND These standards should be viewed as one component of the mathematics education community's response to the current crisis in the teaching and learning of mathematics and the resulting calls for reform. The standards reflect, and are an extension of, the community's responses to those demands for change. Group consensus holds that most students need to learn more, and often different, mathematics and that the teaching of mathematics must be significantly improved. To provide all students an opportunity to learn the mathematics they will need, the emphases and topics of the present curriculum should be altered. More importantly, methods of instruction need to emphasize exploring, investigating, reasoning, and communicating on the part of all students. In particular, teachers should view their role as guiding and helping students to develop their mathematical knowledge and power. Thus, these curriculum standards provide a basis for changing the intent of present courses and for developing different topics and new programs. They characterize the spirit in which instruction should take place. In addition, in order to provide valid evidence that such programs actually improve students' mathematical power, methods of gathering evidence are offered to evaluate both student performance and program effectiveness. THE NEED FOR STANDARDS The driving force for the development of curriculum and evaluation standards is a vision of the mathematics all students should have an opportunity to learn and the way in which instruction should occur. Classrooms should be places where interesting problems are explored using important mathematical ideas. For example, in various classrooms one could expect to see students recording measurements of real objects, collecting information and describing their properties using statistics, and exploring the properties of a function by examining its graph. This vision sees students studying much of the same mathematics currently taught but with quite a different emphasis; it also sees some mathematics being taught which in the past has received little emphasisin schools. Thus, the first aspect of each standard is the mathematical content to be learned. This vision sees classrooms in which students are actively engaged in making conjectures and discussing ideas. Thus, the second aspect of each 24 standard specifies the expected student actions associated with doing mathematics. Finally, this vision sees teachers encouraging students and probing for ideas. Hence, each standard is then elaborated upon in a discussion of the purpose, the emphasis, and the spirit of this vision for instruction. Although many readers may object to this picture of a mathematics class as too idealistic or impossible to accomplish, we are convinced otherwise. The traditions, assumptions, and constraints underlying current educational practice must be changed. In too many schools, teachers find it impossible to teach such mathematical topics or create such classroom environments because of the constraints placed on them. The mathematics presented in many textbooks is a vast collection of vaguely related concepts and skills which are to be mastered in a strict order. Each page is written as if it were independent of all other pages. The only apparent objective is that students become competent at performing a special algorithmic procedure which yields correct answers on sets of stereotypical exercises. And student work is evaluated via paper-and-pencil tests; students' thought processes on reasoning skills are not considered. This fragmentation, the emphasis on paper-and-pencil procedural skills, and a simplistic form of evaluation have effectively separated students from mathematical reality, inquiry, and intellectual growth. Today, teachers also are constrained by directives about chapters or pages to cover, time for instruction, and tests. They lack the necessary resources, the time to reflect, and the opportunities to share ideas with other teachers. Under such conditions it is difficult to create in themselves or in their students a sense of exploration, curiosity, or excitement. Whereas new standards alone cannot alter these conditions, they implicitly argue for such change. We are convinced that many educators are eager to reform school mathematics. During the past decade, numerous attempts have been made to respond to the demands for reform. For example, the majority of state legislatures have mandated that mathematics requirements for graduation be increased and that competency tests be used for student promotion. Similarly, -new syllabi have been or are being developed by school districts and state departments of education. Publishers are producing kits of manipulatives and software and are including more challenging problems in their texts. In addition, a number of experimental development projects and several research studies have been funded by the federal government and private foundations. There is a groundswell of activity aimed at the improvement of school mathematics. Unfortunately, such a piecemeal approach to reform is unlikely to have a significant effect. Evidence from past efforts to change schools makes it clear that a more systemic, orchestrated approach is needed (Romberg and Price, 1981). The NCTM Standards Project was initiated to provide a basis for the development of school mathematics in line with a vision which reflects contemporary mathematics, a contemporary conception of learning and instruction, and the professionals' beliefs about what students should know. Finally, we believe that NCTM is the appropriate agency to develop and disseminate curriculum and evaluation standards. Knowledgeable mathematics educators, through the professional organization which best reflects their interests and those of their students should assume responsibility for the development of standards that closely affect their work and the mathematical learning of their students. For too long we have abdicated this responsibility to others, including legislators, administrators, and text and test publishers. THE NCTM STANDARDS FOR SCHOOL MATHEMATICS The picture of school mathematics from which these standards have been developed is based on our concerns about: • the mathematics that students should learn to be productive citizens throughout their lifetimes; • the methods of instruction which engage students in constructive learning; and • the expectations of our society, which make it imperative that more students have an opportunity to learn more mathematics. THE NEEDED CONTENT FOR SCHOOL MATHEMATICS Contrary to what many people believe, mathematics is a dynamic, , growing, and changing discipline. Davis and Hersh (1981) claim that we are now in a Golden Age of mathematical production, with over half of all mathematics invented since World War II. In fact, they argue that "there are two inexhaustible sources of new mathematical questions. Once source is the development of science and technology, which make ever new demands on mathematics for assistance. The other source is mathematics itself.. .each new, completed result becomes the potential starting point for several new investigations" (p. 25). Given this explosion of knowledge, it is impossible for any one person to know all there is to know. For school programs we must select carefully and emphasize fundamental knowledge needed for contemporary mathematics. We believe that change is necessary because many important mathematics ideas are not reflected in current programs. These standards have been written with this goal and this concern in mind. To illustrate the changes occuring in mathematics and their potential impact on school programs, we have chosen to look at changes in technology and applications. Advances in technology Most current mathematics programs fail to reflect the impact of the technological revolution affecting our society; The availability of low-cost calculators, computers, and related new technology have already dramatically changed the nature of business, industry, government, sciences, and social sciences. Unfortunately, most students are not educated to participate in this new society. Despite the advancements that have brought untold computational and graphical power to our fingertips, in-school hours are spent drilling on computational procedures in arithmetic, algebra, statistics, and even calculus. Yet any step-by-step procedure involving the manipulation of mathematical symbols according to a fixed set of rules can be accomplished on a calculator or computer. Some procedures are simple enough that they are best done mentally or by hand; others are more complex, or take time to work manually and should be done by machine, as they are in the world of work. Based on this reasoning, we have, made the following assumptions: • Calculators are available to all students at all times (a scientific calculator from Grade 7). • A computer is available in every classroom for demonstration purposes, and every student has access to a computer for individual and group work. The new technology not only has made calculations and graphing easier to do, it has changed the very nature of mathematical research. Lynn Steen, a former president of the Mathematical 25 Association of America, stated that "major forces are reshaping the nature of mathematics . Most obvious is the increasing importance of algorithmic processes in mathematical methods" (1986, p. 7). For example, given that computers can perform many calculations quickly, it is no longer adequate to consider one algorithm. Instead, important mathematical research problems are triggered by such questions as -Can you invent an algorithm? Is this the best algorithm? What do we mean by best? How can we prove that it is the best? Since the computer is one force that is changing mathematics, we have assumed that • students should learn to use the computer as a tool for processing information and performing calculations in order to investigate and solve problems. Expanding use of quantitative methods Complementing the influence of technology on mathematics is the fact that the use and application of mathematics have dramactially expanded. Quantitative and logical techniques have permeated almost all intellectual disciplines. Change has been particularly great in the social sciences and the life. sciences. The computer's ability to process large sets of information has made quantification possible in such' areas as business, economics, linguistics, biology, medicine, and sociology. Furthermore, the' fundamental mathematical. ideas needed in these areas are not necessarily those studied in the traditional algebra-geometry-precalculus_calculus sequence. That, sequence was designed with engineering and physical science applications in mind. Since mathematics, is a foundation discipline for other disciplines and grows as a result of its utility, we have assumed that • the curriculum for all students must provide opportunities to build mathematical models, structures, and simulations. within many disciplines. Changes in technology, broadening of the areas in which mathematics is applied, and growth in mathematics itself have changed the problems of importance to mathematics .and the 'methods mathematicians use to investigate problems. These changes must be echoed by changes in the curriculum so that students have an opportunity to learn, the skills and knowledge that are likely to be fundamentally important in. their lives. MATHEMATICAL CONTENT OF THE CURRICULUM STANDARDS We have, used ideas about changes in mathematics to develop curriculum standards by organizing 26 into three categories the content students should have an opportunity to learn. Empirical concepts Initially, mathematical knowledge arose from rudimentary ideas acquired through perceptions of situations in the complex world around us. Several millennia ago our ancestors planted the first seed for this enterprise by observing some quantitative and spatial regularities. From these humble beginnings, mathematics has flowered into the impressive body of knowledge we have been fortunate enough to inherit. Thus, from its origins, mathematics was an empirical science. Its fundamental terms, signs, symbols, and rules are merely abstractions and inventions created to represent properties observed in the environment. Thus, numbers were created to represent the numerosity of sets of familiar objects; signs such as "+" were invented to represent the addition of numbers, which in turn represents the joining of sets, and terms such as parallel and perpendicular were introduced to name spatial properties. The purpose of creating such a language is to communicate with others; the terms of the language become useful only when there are shared meanings. Thus, our first category involves helping students to understand mathematics as a language [we have] created to describe the world and a set of rules for the use of that language. Its origins exist in the regularities of the world in which we live. Furthermore, like any language, mathematics grows and changes as a i-esult of empirical investigations. Thus, all students need to confront a rich array of common problem situations from which the empirical language of mathematics, its notations, and its rules can gradually be built. Abstract concepts A good deal of mathematical knowledge has been created by making the empirically based language and rules of mathematics the objects ofinvestigation. By observing properties of numbers, operations, and spatial figures, man has created another set of terms, signs, symbols, and rules. Some are generalizations of empirical procedures. For ex, ample, the creation of computational algorithms for empirical processes has made mathematics applicable to many seemingly unrelated problem situations. In addition, no' longer bound b,: the concrete, [we have] extended mathematics. by asking "what if. . .?" questions abstracted. from the perceptual reality. For example, while the' 'crea tion of an equation for the distance between two points in a plane has empirical origins, the general ization.of this formula 'to two points in ndimensions does not. And, while multiplication of whole numbers has empirical roots in the grouping of objects, more abstract multiplication algorithms do not. They grew out of observations about properties of exponents such as a X a = + , and the fact that any decimal number can be expressed as an exponent through use of powers of ten. Thus, our second category of mathematics emphasizes the study of abstract systems. Hence, all students should have the opportunity to explore the properties of empirical mathematics and to see for themselves the relationships, rules of transformations, extensions, and structures derived from these investigations. It should be noted, too, that some rules (or algorithms) for transforming symbols into other symbols are so useful that knowledge of them has been considered essential. In fact, mastery of many computational algorithms has been considered by many to constitute mathematical knowledge. We do not deny that knowledge of, in contrast to proficiency with, computational algorithms is important, even at a time when calculators and computers perform most computations for us, but such knowledge should grow out of the problem situations which have given rise to the need for such algorithms. Furthermore, skill in procedural algorithms should not be the backbone of elementary school mathematics. Higher order reasoning skills The intellectual methods that mathematicians use when developing conjectures, reasoning about phenomena, building abstractions, validating assertions, and solving problems are critical to doing mathematics. However, these intellectual skills cut across all content networks. For example, no proposition in mathematics is considered to be true until it has been validated. Initially, justifications may necessarily be built upon empirical evidence since they originated in our perceptions. However, proving an assertion by presenting a rigorous, logical argument has become the hallmark of abstract mathematics. For example, no geometer who had measured the base angles of an isosceles triangle would conclude that they were congruent based on a demonstration, no matter how accurate the measurements. However, such measurements may have formed the basis of a conjecture about their congruence. Mathematicians demand that this result be deduced from the fundamental concepts of geometry. Thus, the discipline grows by applying mathematical methods to a wide variety of problem situations. In this regard, to gain mathematical power, students need to make conjectures, abstract properties and relationships from problem situations, explain their reasoning, follow arguments, validate assertions, and communicate results in a meaningful form. In summary, as the curriculum standards were being written, the mathematical content that students should have an opportunity to learn was foremost in our considerations. THE NEEDED METHOD OF INSTRUCTION The second part of each standard lists important actions of students. These actions are expected to result from engagement in specific instructional activities needed to foster mathematical power. We followed two general principles in stating these actions: first, we have assumed that instructional activities grow out of problem situations; note that while a specific lesson may not be problem based, a sequence of lessons should be grounded in a problem situation. Second, since learning occurs by construction, not absorption, we have described some of the conditions necessary for such learning to happen. Problem situations Traditional teaching emphases on practice in notational transformations and solution algorithms as necessary precursors to coping with a problem ignore the fact that knowledge emerges from the problems. This suggests that instead of the expectation that skill in computation should precede word problems, experience with problems helps develop the ability to compute. Thus, present strategies for teaching mathematics by first teaching skills and then exposing students to stylized application problems need to be reversed; knowledge should emerge from experience with problems. A genuine problem is a situation in which, for the individual or group concerned, one or more appropriately analogous strategies have yet to be developed. The situation should be complex enough to offer challenge, but not so complex as to be co-operatively insoluble by the group of students. In fact, problem situations should parallel the kind of phenomena for which mathematical structures have been typically created (Freudenthal, 1983). Thus, pupils need to experience the phenomena for which such concepts, structures, and ideas were created. In order to teach the mathematizing of situations, teachers need to create suitable contexts; the most abstract mathematics needs the most concrete contexts. Learning as construction In most classrooms, the conception of learning is that students are passive absorbers of 27 information, storing it in easily retrievable fragments as a result of repeated practice and reinforcement. Research findings from psychology indicate that learning does not occur by passive absorption (Resnick, 1986). Instead, individuals approach each new task with prior knowledge, assimilate new information, and construct their own meanings. For example, before young children are taught addition and subtraction, they can already solve most addition and subtraction problems using such routines as "counting on" and counting back" (Romberg & Carpenter, 1986). As instruction proceeds, children often continue to use these routines to solve problems, in spite of being taught more formal procedures. They will .... accept new ideas [only] when their old ideas do not work or are inefficient. Furthermore, ideas are not isolated in memory but are organized and associated with the natural language that one uses and the situations One has encountered in the past. This constructive, as opposed to passive, view of the learning process must be reflected in the way mathematics is taught. Instruction based on this conception of learning is different from that in the typical mathematics classroom where the teacher is transmitting lessons through exposition to a captive audience. Instruction from this perspective should include: • project work rather than exercises; f group assignments as well as independent work; • discussions about the origin of and relationships among concepts, rather than independence of concepts; and • intrinsic motivation through curiosity rather than extrinsic reward. INSTRUCTIONAL ACTIVITIES IN THE CURRICULUM STANDARDS Our ideas about problem situations and learning are stated in the standards. They are reflected in the verbs used to describe student actions (e.g., to investigate, to formulate, to find, to verify). These verbs describe student work on different types of problems. 'Although each problem situation is in one sense unique, we have considered them under seven general headings. Note that, while each category of actions is different in an important way from the others, they are closely related. Examine Many of the activities, particularly when one is introducing new content, involve examining or exploring some situation. The purpose of such activities is to abstract or to invent. To abstract means to identify the intrinsic.form or essence, shape or structure of a given situation. To invent 28 means to produce for the first time through the use of one's imagination, ingenious thinking, or experimentation. Pattern recognition is at the heart of mathematical thinking and invention is so intimately linked with abstraction as to be inseparable. Labels (terms, signs, or symbols) are invented to represent most abstractions. An essential part of these activities is negotiation with students about the choice and purpose of such labels. Represent - Activities of this type expect students to form or to produce an image, picture, word, sentence, chart, or table using appropriate symbols for the referents in a problem situation. These are all part of the process of learning to construct mathematical models of problem situations. Transform A transformation is an operation that involves changing one expression into another following one or more rules. Much textbook mathematics involves learning how to perform transformations. Examples include computational algorithms, division of polynomials, and transforming from one representation to another, such as from an algebraic description of a function to a geometric representation as a graph. Students need to understand why such transformations are desirable and the steps involved in performing the changes, even though rapid proficiency may no longer be essential. Prove When students have made conjectures about a situation, they need to establish evidence that compels acceptance of such propositions. The kind of evidence and form of argument, as mentioned earlier, may vary but the intent.remains constant. Students should be able to construct cogent arguments in support of their claims.. Apply To apply is to put to use known concepts and procedures in new situations. Note that, in applying, one might invent. In the process of inventing, one might apply, 'and; as a precursor to inventing, one might abstract. Solve problems This category refers to true problem-solving situations; i.e., 'those in which complex situations are presented where it is assumed that students have acquired the necessary concepts and procedures to find a solution but need to-use some strategies (heuristics) to make the connections between the given information and a method of solution. Communicate This category of activities involves exchange of information between individuals by use of a common symbol system. Thus, students should be given numerous opportunities to express mathematical ideas verbally and in writing, and to comprehend and interpret ideas expressed by others. • the ability to work with others on problems; • the ability to see the applicability of mathematical ideas to common and complex problems; • preparation for open problem situations, since most real problems are not well formulated; and • belief in the utility and value of mathematics. As students' mathematical power grows and changes, the problem situations must vary with the maturity—both mathematical and cultural— and experience of the students. For example, the primary grades should emphasize the empirical language of the mathematics of whole numbers, common fractions, and descriptive geometry. In the middle grades, the empirical mathematics should be extended to other numbers and the emphasis should shift to building the abstract language of mathematics needed for algebra and other aspects of mathematics. High school mathematics should emphasize functions, their representations and uses, modeling, deductive proofs, and applications. Notice how different this description is from that of a student working alone on sets of well formulated exercises. While mathematics is not taught in schools solely so students can get jobs, their experiences should not be that dissimilar from the expectations of employers. In summary, in our view, instruction should be based on problem situations. As long as the situations are familiar, conceptions are created from objects, events, and relationships in which operations and strategies are well understood. This forms a framework of support which can be drawn upon in the future, when rules may well have been forgotten but the structure of the situation remains embedded in memory, a foundation for reconstruction. Situations should be sufficiently simple to be manageable, but sufficiently complex to provide for diversity in approach. They should be amenable to individual, small-group, or large-group instruction, involve a variety of conceptual domains, and be open as to the methods to be used. SOCIETAL EXPECTATIONS FOR SCHOOL MATHEMATICS In developing the overall set of standards, we were cognizant of other evidence that changes in school mathematics are needed. First, employers have claimed that recent graduates have insufficient knowledge for employment. Henry Pollak (1987), a noted industrial mathematician, recently summarized what industry expects of its new employees, including: • the ability to set up problems with the appropriate operations; • knowledge of a variety of techniques to approach and work on problems; • understanding of the underlying mathematical features of a problem; Second, college and university officials have documented that 15% of the current enrolment in mathematics courses at public four-year colleges is for remedial courses, and another 37 percent is for precalculus courses (Albers, Anderson & Loftsgaarden, 1987). Two probable reasons may explain this situation: first, a mismatch may exist between what is studied in school mathematics and what is needed in some college majors. Second, there has been a change in college expectations: that is, more majors now require mathematics—a fact that many high school counsellors have failed to recognize. Third, the bleak national performance data cannot be ignored. For example, results from the National Assessment of Educational Progress (NAEP) in mathematics (Carpenter, Brown, Kouba, Lindquist, Silver & Swafford, 1978) clearly show that, although most students are reasonably proficient in computational skills, the majority do not understand many basic concepts and are unable to apply the skills they have learned in even simple problem-solving situations. Add to this fact that, when compared with students in other industrialized nations, particularly those in the Orient, our students do not fare well (McKnight, et al., 1987). We expect less of our students, they spend less time studying mathematics, and fewer are enrolled in advanced mathematics than are students in other countries. Fourth, and more disturbing, is the fact that the performance and enrolment picture is even more bleak for women and most minorities. For example, on the average, black students complete approximately one year less high school mathematics than their white classmates (Anick, Carpenter & Smith, 1981). Women and most minorities are seriously . underrepresented in careers utilizing science and technology.. Only 13% of the nation's scientists and engineers are 29 women and only 2% are black (National Science Foundation, 1981). Furthermore, given that schooling practices are inequitable, this condition is likely to get worse. Affluent suburban school districts already provide their students more opportunities and resources for the study of mathematics, and they are likely to be the first to react to the current crisis and recommendations. They are already spending more money on computers and teacher in-service, thus widening further the opportunity gap between affluent suburban students and their poor, urban counterparts. Finally, another aspect of this concern is the belief held by many that the mastery of paper-andpencil procedural skills is necessary before one can investigate problems. In particular, it is a common assumption that arithmetic computations are necessary before one can study algebra or geometry or investigate applied problems. This assumption is not warranted. Too many students are refused an opportunity to learn the mathematics that would make it possible for them to be productive members of society because they ar not proficient at skills which are now done best on a calculator or computer. Even if there were no technological revolution, or explosion of mathematical knowledge, or research on learning and teaching, there is ample evidence that school mathematics would need to be reformed. Taken together, the evidence makes reform imperative. We also have assumed that: • All students should be required to study mathematics for at least 12 years (Grades K-li). • The study of mathematics should revolve around a core curriculum that allows all students an opportunity to learn the important ideas and methods of mathematics. • Mathematics should be studied as an integrated whole so that students understand it as a dynamic discipline and an integral part of our culture. • Doing mathematics involves solving problems. • Communication is an important goal of mathematics instruction. • Mathematics should help build students' abilities to reason logically. • New topics (e.g., data analysis, estimation) must be introduced into the mainstream curriculum. • Mathematics should be taught in a natural context. • Students should be encouraged to create, invent, and participate. • Calculators and computers should be used throughout school mathematics. • Success in paper-and-pencil computation need 30 not be a prerequisite to the study of other mathematics. GOALS FOR STUDENTS Five general goals are reflected in the mathematics curriculum standards across Grade K-12. These we have labeled: (1) becoming a mathematical problem solver, (2) learning to communicate mathematically, (3) learning to reason mathematically, (4) valuing mathematics, and (5) becoming confident in one's ability to do mathematics. These goals imply that students should have numerous and various interrelated experiences which allow them to solve complex problems; to read, write, and discuss mathematics; to conjecture, test, and build arguments about a conjecture's validity; to value the mathematical enterprise, the mathematical habits of mind, and the role of mathematics in human affairs; and to be encouraged to explore, guess, and even make errors so that they gain confidence in their actions. The opportunity for all students to have these kinds of experiences is at the heart of our vision of an exemplary mathematics program. The curriculum should be permeated with these notions to the point that they become commonplace in the lives of students. We are convinced that if students are exposed to the kinds of experiences outlined in the standards, they will be able to develop their mathematical power and will become mathematically literate. Becoming a mathematical problem solver The development of each student's ability to solve problems is essential if he or she is to be a productive citizen. We strongly endorse the first recommendation of An Agenda for Action (National Council of Teachers of Mathematics, 1980): "Problem solving must be the focus of school mathematics" ( p. 2). The initial standard at each of the three levels addresses this goal. Learning to communicate mathematically As described in the introduction, the development of a student's power to use mathematics involves learning the signs, symbols, and terms of mathematics. This is best accomplished in problem situations in which the student has an opportunity to read, write, and discuss ideas in which the use of the language of mathematics is natural. The importance of this goal is reflected in the second standard at each level. Learning to reason mathematically Making conjectures, gathering evidence;- and building an argument to support such notions is fundamental to doing mathematics. In fact, demonstration of good reasoning should be rewarded even more than students' ability to find corrent answers. This goal is reflected in the third standard at each level. Learning to value, mathematics The mathematics curriculum should include numerous and varied experiences related to the cultural, historical, and scientific evolution of mathematics so that students can appreciate the role of mathematics in the development of our contemporary society, and explore relationships between mathematics and the disciplines it serves: physical and life sciences, social sciences, and humanities. Students should learn that mathematics is more than a collection of concepts and skills to be mastered. From its origins, mathematics has been simultaneously a highly esoteric and abstract discipline and the most practical of human endeavors. Throughout its history, practical problems and theoretical pursuits have stimulated one another to such an extent that it would be impossible to disentangle them. Even today, as theoretical mathematics has burgeoned in its diversity and deepened in its complexity and abstraction, it has become more concrete and vital to our technologically oriented society. Our culture has become "mathematized." This goal's intent is to focus attention on the need for student awareness of the interaction between mathematics and the historical situations from which it has developed and the impact that interaction has on our culture and our lives. This goal is not reflected in a single standard at each level but permeates each standard. The importance of coming to value mathematics in- the manner just described has recently been demonstrated by Allan Schoenfeld (in press). Given nonsense problems such as 'There are 125 sheep and 5 dogs in a flock. How old is the shepherd?", students had no trouble deciding he was 25 because that was the most plausible numerical combination of 125 and 5. Schoenfeld argues that students arrived at their answers by following a four-step procedure which is commonly taught: (1) read the problem, (2) select the numbers and the relevant operation, (3) perform the operation, and (4) write down the answer. Furthermore, students have found that this procedure usually works and is rewarded in classrooms. Students may master the formal procedures of mathematics and fail to use them sensibly. Mastery of the procedures of typical classroom mathematics is far different [from] learning to think mathematically. The argument Schoenfeld convincingly makes is that one's knowledge is contextually bound, and that one's understanding of a domain is shaped by the practices of those domains. To devleop mathematical power, one cannot separate mathematics from the cultural, historical, and scientific contexts from which it has evolved. One curricular difficulty is that we have paid inadequate attention to how problems are formulated for students. There is generally very little variety in text formats, and neither the teacher nor the students are expected to vary the bland stereotypes so that interesting problems are posed, discussed, extended, and solved. Cultural contexts surely should be used for this purpose. Only by experiencing the interactions of formal mathematics with the cultural, historical, and scientific contexts will students come to value the discipline as an enormous cultural achievement that is dynamic, growing, and truly important in their lives. Becoming confident in one's own ability The mathematics curriculum should include numerous and varied experiences related to the human aspects of the development of mathematics. This allows students to understand that mathematics is an invention of the human mind so that they learn to trust in their own mathematical thinking. As a result of studying mathematics, students need to view themselves as capable of using their growing mathematical knowledge to make sense of new problem situations in the world around them. This goal is not reflected in any single standard but its spirit is reflected in all the standards. To achieve this goal, there must be a shift of authority for the validation of conjectures, ideas, and solutions from the teacher or the answer key to the students' own mathematical reasoning. One powerful way of helping students see their own human.-capacity to do mathematics and to reason about situations is to infuse the mathematics curriculum with examples of the present and past struggles of humans to describe, classify, compare, contrast, measure, and relate phenomena in the world around them. The study of mathematics msut include the development of the mathematical enterprise, the role of mathematics in human affairs, and the habits of mathematical investigation. Lesson frequently should include information about the problems, ideas, and works that have shaped mathematics. By putting mathematics in its historical and cultural context, students should develop an understanding of how people discover and apply mathematics. 31 Another way to nurture students' confidence is to break down the discontinuity between formal methods and the natural strategies they develop in their daily activities. Several recent studies have indicated that employees show more variability in problem-solving procedures when solving jobrelated activities than do students in mathematics classes (see Schliemann, 1984). This research suggests that problem solving in school has to be taught differently if it is to be useful outside the classroom. Only through such experiences will students come to value mathematics as an important part of their daily lives. Finally, as Davis and Hersh (1981) have stated: "Mathematics has been a human activity for thousands of years. To some extent, everybody is a mathematician and does mathematics consciously. To buy at the market, to measure a strip of wallpaper, or to decorate a ceramic pot with a regular pattern is doing mathematics" (p. xi). School mathematics must enlighten all students as to the truth of this statement. on Mathematical Education (pp. 1-7). Boston: Birkhauser. McKnight, C.C., Crosswhite, F.J., Dossey, J.A., Kifer, E., Swafford, J.O., Travers, K.J. & Cooney, T.J. (1987). The underachieving curriculum: Assessing U.S. school mathematics from an international perspective. Cham- paign, IL: Stipes. National Council of Teachers of Mathematics. (1980). An agenda for action: Recommendations for school mathematics of the 1980's. Reston, VA: The Author. National Science Foundation. (1982). Science indicators, 1982. Washington, DC: U.S. Government Printing Office. Resnick, L.B. (1986, April). Education and learning to think. A special report prepared for the Commission on Behavioral and Social Sciences and Education, National Research Council. Romberg, T.A. & Carpenter, T.P. (1986). Research on teaching and learning mathematics: Two disciplines of scientific inquiry. In M.C. Wittrock (Ed.), Handbook of research on teaching: A project of the American Educational Research Association. Third Edition REFERENCES Albers, D.J., Anderson, R.D. & Loftsgaarden. (1987). Undergraduate programs in the mathematical and computer sciences the 1985-1986 survey. Washington, DC: Mathe- matical Association of America. Anick, C.M.. Carpenter, T.P. & Smith, C. (1981). Minorities and mathematics: Results from the National Assessment of Educational Progress. Mathematics Teacher, 74, 560-566. Carpenter, T.P., Brown, C., Kouba, V., Lindquist, M.M., Silver, E.A. & Swafford, J.O. (1987). Results from the Fourth Mathematics Assessment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics. Davis, P.J. & Hersh, R. (1981). The mathematical experience. Boston: Houghton Mifflin. Freudenthal, H. (1983). Major problems of mathematics education. In M. Zweng, T. Green, H. Pollack & M. Suydam (Eds.), Proceedings of the Fourth International Congress 32 (pp. 850-873). New York: Macmillan. Romberg, T.A. & Price, G.G (1983). Curriculum implementation and staff development as cultural change. In G.A. Griffin (Ed.), Staff development: Eighty-second yearbook of the National Society for the Study of Education (Part II) (pp. 154-184). Chicago: University of Chicago Press. Schliemann, A.D. (1984). Mathematics among carpentry apprentices: Implications for school teaching. In P. Damerow, M.E. Dunkley, B.V. Nebres & B. Werry (Eds.), Mathematics for all. Paris: UNESCO. Schoenfeld, A.H. (in press). Problem solving in context(s). In R. Charles & E. Silver (Eds.), Teaching and evaluating mathematical problem solving. Reston, VA: National Council of Teachers of Mathematics. Steen, L.A. (1986). A time of transition: Mathematics for the middle grades. In Richard Lodholz (Ed.), A change in emphasis (pp. 1-9). Parkway, MO: Parkway School District. MATHEMATICS AND COMPUTERS Functions as Models of Reality Brian Copeland Brian Copeland is mathematics teacher at Templeton Secondary School, in Vancouver. f mathematics education is to regain a I measure of intellectual respectability, it must demonstrate that it is more than some esoteric tautology. It can do this by showing the student now that it can help us to see the world in new and useful ways. The computer seems to me an indispensable tool for overcoming the computational difficulties in applying mathematical functions to model real world phenomena. This article attempts to show how major functions can be used to model reality. Experiments in Functional Composition Given two functions f and g, it is often useful in modelling to use a combination of the two, to get some of the behavior from the one function and some of the behavior from the other. The function f*g takes any argument x and first applies g and then f. Students can use the computer to compare f*g to g*f and to describe the behavior of different functions in ordinary language. This will tap into the existing knowledge base. Most important, attention can be focussed on mathematics as a vehicle for seeing and 34 describing the real world in important, quantitative ways. Exercises like those used here can be open ended, leaving room for considerable student exploration. The computer takes care of calculation; the student takes care of supplying ideas for exploration, describing results, explaining interesting features, and suggesting possible applications. Students can start to reinvent the mathematics they need to learn, generating a sense of ownership, a gradual construction from hunch to finished product and actual perception using the acquired concepts. The examples in Figures 1 and 2 show functions that can be composed to display various features such as changing frequency, varying amplitude, and systematic fluctuations. The computer does the low-level computation, allowing the student to concentrate on the higher level cognitive tasks: comprehension, application, analysis, synthesis, and evaluation. Functions: y=2.71t(sin(x)) y=2.71t(sin(3*x)) ............. X—AXIS:-10. . .10 STEP -5 . STEP a Y-AXT: FIGURE 1 Function: Y=s in(tan(x))*(2.71t(-.1*>)) Ii .fl .11 I It Id I II,' II! I?, Iii, ili .1 ii I IJ I F h1 lit V •I iii ' Oki fl I IIV rV I - till IL X—AXIS: 0.. .50 STEP 5 Y— A XIS: .1 STE P 84 X) FIGURE 2 The Bouncing Ball The bouncing ball (as shown in Figures 3 and 4) needs three functions to model it: • the sine or cosine to get up and down, • absolute value to do away with negative values, • exponential to get the damping effect. After playing with exploratory exercises such as those following, students could model the behavior of specific bouncing balls, with different frequencies, amplitudes, and decay rates. Questions could be set about comparisons with what would happen on the moon and the possibility of a ball that didn't decay but bounced higher and higher each bounce. Preformal experimentation will encourage student independence and confidence. 35 Function: I,' -- V= XR ?—AXIS: 4_P_ ' I 50 STEP 4 . , _L. -J 5. a FIGURE 3 Function: y=7Q*(2.71'D(—.02*'<))*abs(sjn(.Z()*)) !i Ii •F 111111111 l"l i , 311111111 ii ii ii i(, II 1 '.3 111111$ It II I I.' 11 11 1 t• I IIi' •t 1iIIIIIIII.5lIJ'.. . ti 111111) I III'i I sill ' t I F I I I I 111 I I I I I ' ' I I . u irj' '1. I' I'It IIIi i)ii II 1I, 1I.' ('IIfi,i1 I r I ! 1 '1 '11'J I . r .; .• I. .. I,: I ,s •i ('. •., If 111-1 II s '. '.1 C".,. r ? 1 1 ' ''I'Juij . II 1uf'.f.i-....-...-..._. :< - .i • . • '. _ - X—AXIS: 0.. .200 STEP Y — AXIS: 0.. .50 STEP 5 FIGURE 4 Tide Equation Water depth in a tidal region can be modthe definite integral being :done on the elled with the sine function ,(see . igure 5). machine., Amplitude and frequency canbe modelled and discussed in . a way that. connects with existing knowledge., The average value of a Variations on :the basic . exercise, might incontinuous function can, be meaningfully include changing the model to reflect deeper vestigated here (average water de p th ovêra water and seasonal changes ini tidal, depth. tidal cycle). This average value can be A discussion of tides will unpack much of calculated as follows: the information that 'is so. compactly stored ç b f (9' dx in the mathematics. (b - a); 36 Function: y=3*sint(). 5*x ssiss•ilisir', AiiiiIlr' lIItIIi •isiiir•, 1ill55i. .411151'L _i5l5ll5f' iiiiit•. .1.IIillr'. 'iiiti jIlillili .1isssr. I IIIIi ., Illii. j'1Iiil5i. .1 111515i11 ill liii 11111115 15111 sIlls titus StIll 11111 tIll! 111111 stIll lii 1511111 II liii 51111 lillIlsilu 1111115 liii I hut li.iii liii 111111 (1111111 III 15111 IllillilSIl StIll 51151 511111 15111 11111 5111511 111 I 551 1151115111 151111155 5111111 IlIlIIiiIhlliISIIIISlI III liii,,, SIll 11511111111111 Iii III lit 11115151 it slit sill iiii liii iii Still liii I Still 111111 15111 liii! Stilt 511111 StIll Sill still 1 1 i IIl tills 111111 11111 tills (111111 SIll 1151111 iii 11511 liStS l'IIllit Ill III II 111111111 1 11111 111 1111111 111111-1 lli 1 p Hill ItllIIIilIIllIIlillIIIiiillltIIlllIIiIlIIIIitIIillIllliillIihIiIIliIIllittliiiIllhIiIIlilIithIIIIl lIII IlIIl 11111 Iii ill 11111 11111 liii I 111511 11111 liii Ill 11111151 11111 1111111111 111111111111 III 15111 fIlth lIlt lIlt 115111 liii 11111 lilhllhhlI.Iiiiit ilillilil 111111 11111 tills Ills 151111 liii liii ill til 1111111111111111111111111111111111 111151 11111 Hill Hill 111111111111 11111111111111111111 Hill HillI lLJi 11,1111 tJ.jiJ.L_LJJ ILLIJ I LJ.Ji I LJ.JJJ Li.iiJLU ii L11 I Lull Lii it Lii I I 1111111111 Lii it Lii 11111 L Lii I Liii III X) +..}j . s I 'I ( X — AX](S: 0.. .100 STEP 5 0.. 3i STEP I v= FIGURE 5 Motion with Resistance If an object is dropped from a height h above the earth, air resistance ignored: For simplicity, assume v,=O and use v7=g/p to get: v = g/p*exp(_pt) - g/p (1) v = a * t = 32*t d 5 * a * (2 =16*(2 Velocity increases indefinitely with respect to time, and supposedly all objects fall at the same rate, since there is no mention of mass. With air resistance, the situation is different (see equations 1 and 2 below). Some devices even depend on the effects of air resistance (parachute, hang glider, ski jumping). The computer facilitates the investigation of this type of motion under the constraint of resistance, in which: v = (v0 +v)*exp(_pt) -v7 = terminal velocity p = drag coefficient g = acceleration due to gravity Also, the height of the object above ground is given by: y = y0 - vrt + 1/p*(v0+v7)*(1_ exp(-pt)) For simplicity, use v7 =g/p, and assume v.=0 y = y 0_ g/p * t + 1/p*(g/p)*(1_exp(_pt)) (2) Notice in Figure 6 that the velocity does not increase indefinitely, but rather a terminal velocity is attained. Experimentation should then proceed to determine what physical and corresponding mathematical factors will influence terminal velocity. 37 Functions: y(32J0.5)*(2.71f(-0.5*) )-2/O.5 y=(Z32/ 1. 5)*(2. 71t(-1. 5*x) )-32/1 .5 y=(32/.20)*(2.711—.20*))-32/.20 y=(32/.19)*(2.71—.19*))-32,.19 y=(32/.16)*(2.71f—.16*>))-32,.16 X—AXIS: 0.. .50 STEP 5 Y — AXIS : — 260 ... 200 STEP 50 FIGURE 6 The height of objects in free fall can be studied using equation (2). Obviously feathers behave differently from cannon 38 balls. Examples of different free falls are plotted in Figure 7, and the student can explore the different possibilities. Person with parachute p=1.5, V,=21 ft/sec Person with no parachute p=0.15, V,=213 ft/sec Person with unbuttoned overcoat p=0.5, V,'64 ft/sec Funi:t i cns: y=50C)O21.3*+1/1.5*(2l.3)*(:12.71t(t.5*X) ) 15*: :i 71-1% y5000-213.*x+1/. 15* ('213. y=0OO-64.O*x+1/0.5*(64.0)*(1-2.711,(—.50*)) X—AXIS: 0.. .40 STEP 5 STVP Y-AT: FIGURE 7 Damped Oscillatory Motion Damped oscillatory motion refers to the progressive reduction or suppression of the oscillation of a system. Vibrating springs and pendulums gradually fading out are both examples of decaying periodic behavior. Modelling periodic phenomena with decreasing amplitudes can be accomplished using a trigonometric function t and an exponential function exp(-bx). This is of the form A*t(x)*exp(_bx) (see Figure 8). Students can study the examples provided and attempt to model an actual spring or pendulum in oscillation, by adjusting parameters in the general function. At this point in their development, students should be able to research applications of this type of functional variation. They should also be able to generate reverse damping and suggest applications, e.g., destructive vibration amplification, sound theory. 39 Fun ';ti ':'ns: y=2*sin(2*x-3. 14) y=4*si n (2*'-3. 14) y8*sin(4*x-3. 14) y=9*sin(1*x-3. 14) *(2.71•Dc—x/3:)) *(2. 71t(—x/5)) X—AXIS : 0.. .15 STEP '—AX.IS: .10 STEP FIGURE 8 Uninhibited-Growth Model If a population reproduces itself at a rate proportional to its size, then the population P is given by P(t) = P0 exp(kt), where k is the % growth per year as a decimal fraction. 40 An investigation of future world population under the dual assumptions of uninhibited growth and percentage increases ranging from 0.5% to 3% is'shown in-Figure 9. Functions: y5*2. 7lt O1O*x) y*2. 71tC . 015*x) y*2. 71t( .020*x) y*2. 71i'( ..030*x) y=*2. 71 -t .005*x) y=5*2. 71 -r O(-.)o*x:) X—AXIS: — 100 ... 100 STEP 10 Y—AXIS: 0. .50 STEP 5 FIGURE 9 Exponential Decay If a population declines at a rate proportional to the present population, then P(t) = P0exp(-kt), where k is the % yearly decline as a decimal. The constant k for carbon-14 is .00012. The table of values and graph in Figure 10 can be used to show, for example, that the half life of the isotope is something like 5750 years and that if only 70% of an object's C-14 remains, the object is about 3000 years old. 41 x 100*(2.71t(-.00012*x 0. 1000. 2000. 3000. 4000. • I I ri . 5000. 6000. 7000. 8000. 9000. 10000. 11000. 12000. 13000. 14000. 15000. 16000. 17000. .18000. 19000. 20000. • 100. 88.72 78.72 69.84 61.97 54.98 48.78 43.28 38.4 34.07 30.23 26.82 23.8 21.11 18.73 16.62 14.75 13.08 11.61 10.3 9.14 1 -.ri - •-• - . y=1(:)0*(.2.711'(-.00012*x)) y=70 ---hX]tS: •. . . 20 1000 STED X Y -Am 1rs: b •-i41 STFP FIGURE 10 Inhibited-Growth Model If a population is constrained by certain limits to growth such that dP = kP(L-P), Ji- 42 where L is the limit and k the growth rate, then the population is given by p*L P(t) - P0+exp(_Lkt)*(L_P0) The inflection point of the curve, where the rate of change reaches a maximum and then declines, is P = 0.5*L. The graph and table of values in Figure 11 depict world population under the contraint of L=30 billion. An informed discussion of the issue could be carried on using the model as a quantitative basis. Fun': t j csns: y=(5*30)/(5+2.71i(30*.01*x)*(303)) ,= : 5*30: / (5+2. 71 t (-30*. 001 *x) * ( 30-5)) y=(Z*3o)/(5+2.71i(-30*.O03*x)*(3o-5):) y(5*30)/(5+2.71D(-30*.020*x)*(30-5)) ---.-•1 / f .11 j• ,' • 1 ___:------------ . X — AXIS: B...50 STEP 5 Y—ATS: A- 4PJ STEP Figure 11 continues . . 43 (5*30) / '5-'-2. 71i(-3c:..o2o*:*:3o-5: :) '.J. 14.15 23.98 28.4 29. 3 29.92 f3 S.. -. 5. 7.5 10 125 15. 17 2. 22.5 1=1 30. 30. 30. 30. 3o. 30 4.. 5 43- 47. 30. 0. FIGURE 11 The computer can make the meaningful application of mathematics a reality for more students than ever before. B.C. still lacks a program to take advantage of the powers of 44 the computer to help students master the fundamentals of quantitative literacy. This failure remains a chronic problem in B.C. mathematics education. Secondary School Calculus Harold Brochmann Harold Brochmann is a computer consultant with the North Vancouver School District. This is the first in a series of three articles about teaching calculus in secondary schools. n their first year of college or university, I many students take one or more courses in calculus. The courses are quite difficult, and many students do a great deal of work to get through. Many students fail. Getting a decent grade in first-year calculus matters not only to the students, but also to their secondary school mathematics teachers. Several post-secondary institutions make a regular practice of releasing the results of the first-year exams, broken down by school attended. This practice, together with declining enrolment, professional pride, and plain need to cover one's butt, prompts secondary math teachers to go to great lengths to assure their students' success in the year after they part company. Enter Calculus 12. Calculus 12 is offered as an "elective" in many school districts, and those that don't offer it yet are actively preparing for the inevitability. Privately, teachers will agree that if the "competition" didn't offer such a course, there would be little pedagogic justification for doing so, but "under the circumstances, we have no choice." Ah yes, calculus. I spent several years differentiating and integrating without the slightest idea of what it was all about. Calculus, like so many topics in mathematics, is usually presented to the student as a ritual to be performed according to rules laid down by person or persons unknown for purposes also unknown, or at least not to be revealed just yet. My Grade 9s used to ask, "Why do we have to learn algebra anyway?" The answer was "Until you have learned it, you won't be able to understand the reason for learning it. Therefore it's no use trying to explain." Are the reasons for doing calculus the way we do it any better? What is calculus anyway? I've finally figured it out. Calculus is a collection of symbol-manipulating algorithms, techniques, tricks, rules, and shortcuts that allows one to find slopes of, and areas under, curves. Do I hear any objections? Now that we have a definition, we are left with the questions of which should be taught 45 first: • the traditional collection of symbolmanipulating algorithms, etc.? • the conceptual basis from which the algorithms are derived? • what you can do with them? My vote is for the last of these. The first, as far as I'm concerned, should be taught last, and not as the core content of a first course. The concepts of limits, like all concepts, cannot effectively be taught, but will form in students as a result of using them in applications. Now the question arises as to whether it is possible to learn about applications of calculus without first knowing the contents of traditional calculus courses. Is it possible to acquire the concept of square root and learn to solve problems involving square roots without knowing one of the algorithms for finding it? I say yes. Using a calculator to solve application problems surely leads to sound conceptual understanding. I suspect that few teachers bother with the algorithmic treatment at all. How many of us could calculate the sine of 74 degrees from first principles? Does this 46 stop us from teaching trigonometry? How many of us who did lengthy numerical calculations using logarithm tables can find loge 526.7 with a pencil and paper? Can you find 866.45 without your calculator? I can't. So, why not teach calculus applications with a computer—at least as a prelude to the traditional treatment for the purpose of providing conceptual grounding? I submit that this is a more appropriate course content for Calculus 12 than attempting to pre-teach the material the students will get next year. This paper is not an exhaustive course description. It illustrates a general approach to a unit that should, in my opinion, be taught prior to a traditional calculus course. There are three articles in this Vector series: 1. Introduction to integration using the wellknown computer language BASIC. 2. Applications of integration using computer utility programs. 3. Computer-based differential calculus with applications. V What Is Numerical Integration? 10 This is the graph of Y = X 2over the range 0 <'X <10. Our purpose is to determine the area between the curve and the X axis. 8 7 6 $ 4 3 2 0 0 5 6 7 12345 6 7 8 6 7 8 9 10 I 2 3 4 9 10 9 10 x V The midpoint of the rectangle is at X = (1/210 The height of the rectangle at the midpoint is Y = [((1/2)*10)-21/10 A = 10*[((1/2)*10)^21/10 =25 If we make two rectangles, then their midpoints are at '((1/2)/2). *10 and Xi •X2 = ( ( 3/2)/2)10 The, heights' are = [(( ( 1/2)/2)*10y2]/10 Y2 = [(( ( 3/2)/2)*10)A2]/10 Y1 10 9 8 7 6 5 4 0 0 V 10 9 8 7 6 5 4 Combined area of both rectangles is A = 10/2*f[(((1/2)/2)*10)2}/10 + [(((3/2)/2)*10y2]/101 = 31.25 X 47 With three rectangles, we get midpoints at V xl = ((1/2)/3)*10, X 2 = ( ( 3/2)/3)*10 and X3 = ( ( 5/2)/3)*10 9 The heights of these rectangles are 7 10 8 6 Yl = [(((1/2)/3)*10y21/10, Y2 = [(( ( 3/2)/3)*10)A2}/10 and Y3 = [(( ( 5/2)/3)*10y21/10 5 4 3 I I I I 2 This gives a combined area of A = 10/3*1[(((1/2)/3*10)^2]/10 • [(((3/2)/3*10)-2]/10 • [(((5/2)/3*10)-2]/101 = 32.407404 With four rectangles, we get midpoints at xl = ((1/2)/4)*10, X2 = ( ( 3/2)/4)*10, X3 = (( / 2)/4)*10 and X4 = ((7/2)/4)'10 The heights of these rectangles are V 10 9 8 7 Yl = [(((1/2)/4*10)^2]/10, Y2 = [(( ( 3/2)/4*10)A2]/10, Y3 = [(( ( 5/2)/4*10)^2]/10 and Y4 = [(((7/2)/4*10)A21/10, This gives a combined area of A = 10/4*t[(((1/2)/4)*10y21/10 + [(((3/2)/4)*10Y21/10 + [(((5/2)/4)*10)A2]/10 + [(((7/2)/4)*10)2]/10 -. 32.8125 48 6 5 4 3 2 0 0 I 2 3 4 5 6 7 _x 8 9 10 In general, the sum of the areas for n rectangles is A = ( 1O/n)*t[(((1/2)/n)*10Y2]/1O + [(((3/2)/n)*10Y21/10 + [(((5/2)/n)*10)2}/10 + [(((7/2)/n)*10)A2]/10 + [(((9/2)/n)*10Y21/10 + [((((2*n_1)/2)/n)*10)2]/101 We now make three statements: 1. The sum of the areas of the rectangles we have drawn is very approximately equal to the area under the curve. 2. The more rectangles we use, the closer the sum of the areas will be to the area under the curve. 3. If we use a sufficiently large number of rectangles, then the sum of their areas will be close enough to the area under the curve. How many rectangles are needed in order for the results to be close enough? The answer to this question is determined-by using a computer program that finds the sum of the areas of 10, 100, 1000, and 10 000 such rectangles. The program is written in BASIC and looks like this: 10 A0 : N=10 20 FOR Z=1 TO 2*N1 STEP 2 30 A =A +((((2*Z1)/2)/n*1O)2)/1O 40 NEXT Z 50 A=A*10/N 60 PRINT A Running this program yields an area of 33.25. Changing line 10 to N=100 gives 33.3325. Using N = 1000 gives 33.33251, and with N=10 000 we get 33.333333. As it happens, the area under the curve is theoretically 33.3.... With 100 rectangles, we get four digits of accuracy, which is adequate for most purposes. In this example, we found the area under the curve Y = . Generalizing this to Y = f(X), the sum of the rectangles that approximate the area under the curve for the range 49 70 NEXT 80 A=A*(MXMN)/N 0< X <10 becomes A = 10/n*tf(((1/2)/n)*10) • f(((3/2)/n)*10) • f(((5/2)/n)*10) • f(((7/2)/n)*10) 90 PRINT A Line 30 in the program defines the function used in line 60. To use the program for other curves and/or for other ranges of X make changes to line 20 and 30. + f[(((2*n_1)/2)/n)*10)1 Also in this example, we used the range 0<X<10. Generalizing this for range Xmin<X<Xmax, we get Integral calculus is the branch of mathematics that deals with finding areas under curves. What we have been doing here is numerical integration. The conventional mathematical notation for the last example is A = ((XminXmax)/fl *[f(((1/2)/fl)*(Vmax Xmin) +X min ) ' + f(((3/2)/n)*(X max Xmin)+Xmin) + f(((5/2)/n)*(X ' max Xmin)+Xmin) ± f(((.7/2)/n)*(X max Xmin)±Xmin) f[(((2*n_1)/2)/n)* ( Xmax - Xmin) + XminIl The final version of the computer program which calculates the approximate area under any curve Y=f(X), for example Y=3*Xfr52_7 over any range, for example, 5< X <17 now becomes 10 A0 : N1000 20 MX10 : MN0 30 DEFFNA(X)=3*XS2_7 40 FOR Z=1 TO 2*N_1 STEP 2 50 X=Z/2/N*(MXMN)4-MN 60 A=A+FNA(X) 50 S (3x7)dx Note that the. BASIC program and the con-: ventional mathematical notation (CMN) are two different ways of expressing (for all practical intents and purposes) the same thing. The CMN version represents the theoretical limit to the area under the curve, while the other actually calculates the area to some (number of digits of) accuracy depending on the computing facilties used, and the precision. 5specified. The program finds a solution quickly because it is machine executable and it is adaptable to any function. Integration by conventional techniques always requires a great deal of training, is at best time-consuming, and for many functions, extremely difficult.' Applying the Computer Program to Familiar Examples Example 1 We will practise using our computer program to find some areas that we are already familiar with. Consider first the area of this figure: Example 2 The equation for the graph of a circle is x2 + y2 = r2 from which we get y= Consider now this quarter circle (0,5 ) 0 (2,0) (7,0) We know that this area is 5*4 + 5*2/2 = 25. Substituting the values from the diagram into the computer program. 10 A0 : N1000 20 MX=7:MN=2 30 DEFFNA(X)=2/5*X+16/5 40 FOR Z=1 TO 2*N_1 STEP 2 50 X=Z/2/N*(MXMN)+MN 60 A=A+FNA(X) 70 NEXT Z 80 A=A*(MXMN)/N 90 PRINT A Running this program also yields 25. The conventional mathematical notation for what we have done is (2/5 * x + 16/5)dx = 25 [ii (5,0) X To find the area under the curve, the first three lines of the computer program become 10 A0 : N100 20 MX5:MNO 30 DEFFNA(X) = SQR(52-X2) This will yield the area of a quarter circle. To find the area of the complete circle, we change line 80 to read 80 A=A*(MXMN)/N * 4 Running the program produces 78.5484. This compares to * 52 = 78.5398. 51 The slight discrepancies observed are related to compromises made in the computer's arithmetic algorithms. what we have done is 52_x2 = 78.5 The conventional mathematics notation for Example 3 Next, we find the area bounded by a SINE CURVE I 2ir 0 360 With 10 A0 N100 20 MX2*ir : MNO 30 DEFFNA(X) = SIN(X) 80 A=A+(MX—MN)/N the program produces an area of zero. This is because the area below the curve is considered negative. With 20 MXir:MNO we get 2. The conventional mathematics notation for what we have done is (sin x)dx = 2 52 Example 4 Find 02 (x/(22+1)3)dx On the right is the graph plotted over the range -2<X<'2 and 0< Y < 100. -2 2 Using the computer program, we find that the numerical solution is 0.111111.... -di The purpose of integration is to find the area under a curve. Although this area can be expressed symbolically, it is the numerical value which is needed in applications—and it can be found quickly to any needed degree of precision using a simple, virtually intuitive, "homemade" computer program. It is always desirable to have available the graphs of the functions we integrate. In the first three examples, we saw, the graphs of the functions were familiar, and plotting the graphs is quite straightforward. Plotting the graph of the last function involves some pretty tedious calculations. In the next article of this series, we have a look at another "homemade" computer program that not only performs the calculations, but also draws the graphs for us. We also look at a commercial program that determines appropriate "best fit" functions from raw data. 53 MISCELLANEOUS Report on the 27th Northwest • Mathematics Conference "Mathematics for the '90s" Ian 'deGroot Ian deGroot is president of the British Columbia Association of Mathematics Teachers; he teaches in North Vancouver. he Empress Hotel in Victoria was a T gracious hostess to more than two thousand mathematics educators from Oregon, Washington, and British Columbia. It was great to renew friendships with our colleagues and to welcome distinguished visitors such as Shirley Frye, president of the National Council of Teachers of Mathematics.' The organizing committee is to be commended for the superb program and smooth organization. Everything went off without a hitch. We tip our hats to Bob Belcher, Trevor Calkins, Brent Carberry, Dana Close, John Clemens, Joan Foster, Liz Hamblett, Lyle Melnyck, Betty Milne, Daphne Morris, Wendy Swonnell, Brian Tetlow, and Jim Vance. A standing-room-only audience in the Empress Ballroom greeted keynote speaker Frank Ebos as he provided his insights into the teaching and learning process. I knew then that it was going to be the best Northwest ever! Friday's sessions continued in the same positive manner. Lola May, no stranger to 54 B.C. math teachers, was welcomed by a packed audience in Newcombe Auditorium. She held the crowd in her hands with her particular brand of energy and charm as she talked. about calculator 'and computer activities. Richard Skemp, of Coventry, England, presented his view on the teaching of mathematics, and Shirley Frye, of the NCTM, enlightened us with her plan for the NCTM standards for school mathematics. That document will certainly, change mathematics instruction in North America.. I believe that we will be hearing about and experiencing the objectives of these standards over the next few years. Brendan Kelly was his usual dynamic self, with his "New School Mathematics Programs for the 1990s." Frank Ebos was inventive when he had to be as he made his second contribution with "Motivation in Mathematics." No Northwest conference would be complete without the presence of Father Stanley Bezuska, of Boston College. His topic, "Mathematics for the Countdown to the Twenty-First Century," was the icing on the cake for many of tHe participants. Harold Jacobs, the teacher's teacher, presented "Mathematical Snapshots of 1988" to another responsive audience, focussing on the conference theme of Mathematics for the Nineties. We should not forget the usual array of talented local presenters: Jim Sherrill, John Kiassen, Tom O'Shea, Zoe Wakelin, Becky Matthews, Grace Fraser, Don Vance, Ivan Johnson, Jim Nakamoto, Dennis Hamaguchi, Les Dukowski, Bill Kokoskin, Jim Mennie, Bill Toth, Doug Owens, Mark Mahovlich, Tom Schroeder, David Robitáille, Craig Newell, -Alan Taylor, Gary Phillips; David Parkinson, Jim Swift, and Walter Szetela: Have I missed anyone? These dedicated- mathematics educators have shared their ideas and expertise with us for many conferences and have helped to make Our British Columbia organization one of the best in Canada. I was particularly impressed with the two sessions that I attended on the graphing calculators. Jim McManüs and Bob Hill, of Albany, Oregon gave us hands-on experience On the use of the HP 28S symbol manipulator, a truly incredible instrument; I call mine DEBE, an acronym for does •--,•.• everything but eat. In the second session that I attended, Duane Olson, of Lacey, Washington, showed how the CASIO 7000 g graphing calculator will eventually be a necessity in every senior mathematics class. I was delighted with the level of debate at the BCAMT annual general meeting, which was the final session on Friday afternoon. We then all adjourned to sample the hospitality of various publishers' suites. Saturday morning, it was my turn to present, and I chose the topic "Motivation, the Key to Success in Applied Math." Dr. Martin Collis,, -of the University of Victoria, closed the conference with his fitting talk on the unchanging qualities a teacher needs-to teach in a changing world. I left Victoria with many memories of the conference: I won't forget Eldon Egbers, the supervisor of mathematics from Washington State, who told me that he -has attended all 27 Northwest conferences! I won't forget the participation of the publishers and their appropriate displays. I will -not forget the warmth and enthusiasm of all of the participants. I can hardly wait for the Seattle conference in October 1989. : 55 The SFU Master's Program in Secondary School Mathematics Education Tom O'Shea Tom O'Shea is an associate professor in SFU's Faculty of Education. t the present time, there appears to be A renewed interest among secondary mathematics teachers in a graduate program structured to meet their needs. Secondary mathematics teachers are facing a revised curriculum that will be fully in place in 1990. This curriculum will contain a renewed emphasis on geometry, a new strand on probability and statistics, and a new unit on calculus. Our graduate program is designed to help teachers develop insights into the nature of mathematics and its place in the school curriculum. Teachers will also become familiar with research on how secondary students learn mathematics, and current ideas on how best to teach the subject. This is an opportune time for teachers to examine curriculum changes and upgrade their qualifications. The theme of the graduate program emphasizes the human aspects of mathematics: the role of mathematics in society and the natural development of mathematics as a growing, changing, entity. Developments in the school mathematics curriculum, and in pedagogy, will be related to historical, cultural, and psychological forces operating within society. The goal is to produce teachers who have a broad understanding of mathematics and mathematics education, and who will be qualified to deal with rapid curriculum change in the next several decades. To this end, the Faculty of Education and the Mathematics Department at Simon Fraser University have collaborated to develop five courses specifically designed for secondary mathematics teachers. 56 Course Structure of the Program Students require at least 23 credits and a thesis for the Master of Science (Education) degree. Equal numbers of courses will be taken in the Mathematics Department and in the Faculty of Education. The Mathematics Department has developed three new courses for the program, and two new courses have been developed in the Faculty of Education. All courses are four credits. The program has been designed so that teachers can take one course each semester at the same time as they continue their normal teaching career. Classes will be held from 16:30 to 20:30 one day a week for 13 weeks each semester. Classes will be held at SFU's Burnaby campus or perhaps at the Downtown campus, depending on where the majority of students live. At the present time, the program is feasible only for those students within commuting distance of these two locations. Courses offered by the Mathematics Department 1. MATH 601-4 Foundations of Mathematics "Crises in mathematics, their historical and philosophical background, and their resolution." In this course, all necessary mathematics will be taught as part of the course. The intent is to show mathematics in the making rather than as a finished product. Various critical periods in the development of mathematics will be examined. The emphasis will be on the mathematical problem of the time and how the problem was resolved. 2. MATH 602-4 Geometry "Euclidean and non-Euclidean geometries. Klein's Erlanger program." In this course, students will look at the development of geometry to the present time. Emphasis will be placed on how geometry was interpreted at various times in history, including the influence of Euclidean geometry on philosophy, and the crisis precipitated by the discovery of non-Euclidean geometry. Modern geometrical treatment will include transformations of the plane. the teacher." Students will examine implications for instruction of the ideas of various mathematics educators and school of thought, for example, Dienes, Gattegno, Skemp, and the constructivist school. In part, the course will focus on teaching geometry, anticipating the content of the geometry course in mathematics, and on applications and problem solving reflecting the content of the mathematical modelling course. 3. One elective in the Faculty of Education. 3. MATH 603-4 Mathematical Modelling "Introduction to mathematical modelling using algebraic and geometric techniques, along with techniques using calculus." This course is designed to give students experience in creating and fitting mathematical models to real-world problems. It is based on recommendations of the MAA's committee on the Undergraduate Program in Mathematics, and includes modelling using the computer program Minitab. Courses offered by the Faculty of Education 1. EDUC 846-4 Foundations of Mathematics Education "An examination of historical, cultural, and psychological forces shaping the secondary school mathematics curriculum. Current developments in mathematics curriculum and in mathematics education research." The emphasis will be on the historical underpinning of the curriculum and the cyclical nature of reform in mathematics education. The course will have a structure similar to the one on the foundations of mathematics and will focus on critical periods in the development of the school mathematics curriculum. 2. EDUC 647-4 Teaching and Learning Mathematics "The theory and practice of mathematics teaching at the secondary level. Emphasis on the nature of the learner and the function of Prerequisites The program is intended to meet the needs of practising secondary mathematics teachers. The mathematical content is designed to increase students' breadth of understanding, and will not require students to recall details of their undergraduate mathematics courses. We recognize that teachersinterested in this program may have completed their university work some time ago, and the courses are designed with this in mind. In addition to the standard requirements for entry to the SFU graduate program, the following prerequisites are normally required: • sufficient mathematics background for certification to teach secondary school mathematics • mathematics teaching experience in secondary school • an undergraduate methods of teaching mathematics course, for example, SFU's EDUC 475-4 Designs for Learning: Mathematics. Timetable for Implementation The program is designed for a cohort of students. The entire program is a cooperative venture between mathematics and education, and each course is designed to complement the other courses. The initial intake will be in the fall of 1989, with a second intake planned for two years after that time. This schedule may change, subject to demand. Fall 1989 MATH 601-4: Foundations of Mathematics supervision of students' theses. Students will be encouraged to undertake work in which they can apply the ideas from the courses to curriculum development. The deadline for application to enter in fall 1989, is March 1, 1989. For further information, please contact: Dr. Tom O'Shea Faculty of Education Simon Fraser University Burnaby, BC V5A 1S6 (604) 291-3395 Spring 1990 EDUC 847-4: Teaching and Learning Mathematics Summer 1990 MATH 602-4: Geometry Fall 1990 EDUC 846-4: Foundations of Mathematics Education or SPRING 1991 MATH 603-4: Mathematical Modeling Dr. Harvey Gerber Department of Mathematics and Statistics Simon Fraser University Burnaby, BC V5A 1S6 (604) 291-3331 Summer 1991 Education: One elective course For application forms, please write directly to It is expected that students will take one year to complete a thesis. Members of the Facul ty of Education and the Mathematics Department will share responsibility for 58 Graduate Programs Office Faculty of Education Simon Fraser University Burnaby, BC V5A 1S6 (604) 291-4787 Y88-0145 November 1988 utfe National Council of Teachers of Mathematics_______ Proclamation Whereas, mathematical literacy is essential for citizens to function effectively in society; and, Whereas, mathematics is used every day—both in the home and in the workplace; and, Whereas, the language and processes of mathematics are basic to all other disciplines; and, Whereas, our expanding technologically based society demands increased awareness and competence in mathematics; and, Whereas, school curricula in mathematics provide the foundation for meeting the above needs; Now, therefore, I, Shirley M. Frye, President of the National Council of Teachers of Mathematics, do hereby proclaim the month of April 1989 as Mathematics Education Month To be observed in schools and communities in recognizing the increased importance of mathematics in our lives. In witness thereof, I have hereunto set my hand and caused the corporate seal of the National Council of Teachers of Mathematics to be affixed on this 1st day of September 1988. 'J^ President X"TH* You can count on it A special new video designed to encourage students to get excited about math. This fast-paced video shows students enthusiastically dealing with everyday math problems and having lots of fun at it. The video can be purchased by schools and school groups for the cost of duplication—just $8.75 pet copy. If you would like to order this video, please write: VCA Teletronics, 1801 Royal Lane, Suite 1010 Dallas, TX 75229. 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