Chapter 5: Case 2: Randy Walker: Tentative Steps

along the Reform Path

 

In 1983, at a local teachers' conference, Randy Walker was introduced to the subject of fractal geometry. Reading, independent study, and mathematical investigations in this topic have consumed much of his spare time since then and have contributed to an expanded vision of mathematics. The open-ended questions he has met and the experiments that he has performed in the search for answers have excited Randy and convinced him that school mathematics could be much more enjoyable if it broke from tradition. Now he has as a goal a less formal and more student centred program where pupils are engaged in investigations rather than routine symbol manipulation (RW-INT-D-05b). Randy has, to a limited extent, put his plans in place, but during the period of this study there remained many ideas, in his head and on paper, that did not reached the classroom.

When given the course time to pursue his new mathematical interests, Randy engages his pupils in projects and investigations, often computer supported, that involve random events, chaos, and dramatic mathematical art. Students meet significant questions concerning probabilities, make conjectures, debate their reasoning, conduct experiments, and through the observation of patterns build toward, often tentative, solutions. These activities are frequently linked to practical problems such as those of industrial quality control (RW-LES-N-25) or the study of nature's repeating patterns (RW-DIS-N-25). In this environment the mathematics education reform themes of problem solving, communication, reasoning, connections and the use of technology are in play. Assessment activities are expanded beyond the traditional tests and examinations, with students receiving grades for their reports on open-ended investigations.

Randy has focused his efforts in developing an alternative mathematics curriculum and teaching style on the new topics of fractal geometry and chaos. When it comes to teaching the regular school mathematics content his approach is much more traditional. During the time of this study, Randy was working in a school that did not value new mathematical topics and confined itself to the narrow curriculum as described in the content listings of the official guidelines (Ontario Ministry of Education, 1985). Lacking administrative support for the teaching of his new interests, Randy's opportunities to address these topics were pushed to the edges of his program: optional bonus projects for students to complete outside of class, short compressed lessons when some class time remained after the completion of traditional lessons, classes in a loosely organized experimental science course, and extracurricular activities with interested students.

Eighteen of the twenty-three lessons observed during this study were with two Grade 12 classes in the university preparation Academic stream. Here, for the well established topics of trigonometry and exponents, Randy employed almost exclusively a teacher centred mode of instruction. For parts of three of these lessons group activities were organized, but for the most part instruction took the form of Socratic lessons followed by student practice, either individually or in ad hoc groups. With Randy's very detailed explanations and his insistence that pupils give fully reasoned answers, mathematics was presented as a logical activity, but the final outcome of most lessons appeared to be well understood, rules, facts, and procedures.

When listening to Randy outline his conception of mathematics the reader will find that this direct instruction style of teaching is not incompatible with his vision. When addressing newer mathematical topics such as fractal geometry, dynamical systems, and chaos theory, Randy's position has fallibilist trends but generally he views the bulk of the discipline in absolutist terms. Here he presents a blend of Platonist, formalist, and to a limited extent instrumentalist philosophies.

 

Three Coins

 

During the academic year prior to the commencement of my research project I visited Golden District Secondary School to meet with Randy and explore the possibilities of his joining the study. My brief time in Randy's classroom that day provided a vivid illustration of a key theme in his conception of mathematics; mathematical knowledge can be advanced through inductive processes. Sometimes mathematics is an experimental science.

Period one in the day's schedule is already half over when I arrive at Golden District Secondary School, so I proceed directly to Randy's classroom. This is the scheduled time for his special course, "Fractal Geometry and Chaos Theory", a non-guideline course that he has just this year been granted permission to teach. The students are busy working in groups and Randy is circulating around the room, dropping in on each cluster of pupils, asking for reports of their work and posing questions. There is a computer at one side of the room and it is obviously performing some repeated calculations as the same message with different numbers keeps flashing on the screen.

Life of latest coin trio = 2 toss(es)

Average trio life = 527248/167700 = 3.143995 tosses

Life of latest coin trio = 1 toss(es)

Average trio life = 527568/167800 = 3.143981 tosses

Life of latest coin trio = 3 toss(es)

Average trio life = 527885/167900 = 3.144044 tosses

When he sees me at the door, Randy strides over, gives me his usual strong hand shake, asks one quick question about my trip, and then turns the conversation to mathematics. He pulls three pennies from his pocket and announces, "We are going to do an experiment." Randy tosses the three coins into the air so that they fall on the top of the bookshelf near the computer. The coins come up two heads and one tails. "We remove the heads", Randy continues as he sets the two coins aside and picks up the remaining tails. He tosses the coin and it comes up tails again. Randy gives it another toss. Heads results and Randy puts the last coin with the first two. "All three are gone now." "This time it took three tosses for all the coins to come up heads. On average how long will it take?"

Randy's puzzles are always interesting but never easy. I think about the problem but can only give a range for one trial. "It could happen on the first toss but it could also take an infinite number of tosses." "Yes", Randy replies, "But on average?"

Randy tells me how the class has been working on coin toss probability problems and how he modified a situation presented in one of his reference books (RW-LES-D-i, Paulos, 1991) to get this new puzzle. "We have worked on it experimentally and then I wrote this computer simulation - look at the numbers coming up for the average."

Life of latest coin trio = 7 toss(es)

Average trio life = 529404/168400 = 3.143729 tosses

Life of latest coin trio = 1 toss(es)

Average trio life = 529703/168500 = 3.143638 tosses

Life of latest coin trio = 4 toss(es)

Average trio life = 529994/168600 = 3.143499 tosses

"Notice anything interesting?", Randy asks. I note the results are close to 3, actually close to the decimal expansion for B. Randy is obviously excited. "Yes! We keep getting answers close to, but I can not see any reason for this. I'm working on a way to calculate an answer directly, but have not figured it out yet." (RW-DOC-D-02a)

For Randy, mathematics can be explored through experiments. He has no difficulties with his present lack of a definitive answer. This is just another challenge on which to work. I wander around the room and talk to the students as they puzzle over related coin toss problems. They seem to share their teacher's view that mathematics is an experimental science and appear happy to be involved in open-ended explorations. (RW-LES-N-i)

 

Mathematics an Experimental Science

 

Recently Randy has been studying the new mathematical sub-discipline of fractal geometry. Ideas from his reading and courses that he has taken at a local university appear throughout the materials he produced in this study's structured activities: the writing on the nature of mathematics, repertory grids, and his concept map for mathematics; and resurface regularly in subsequent interviews examining these works.

In Randy's eyes the nature of mathematics is changing and in discussing the sources of mathematical knowledge he writes, "Computer experiments now drive much of mathematics research" (RW-DOC-D-07). Later, in an interview, Randy acknowledges the source of his new vision of mathematics. "I probably never would have said that if I hadn't been involved in fractals over the last few years. I never would have had a notion of mathematics as being very experimental" (RW-INT-D-01b).

Randy recognizes that his view of mathematics as an empirical discipline is not universally popular within the mathematics community, but he sees progress on this point.

Some [mathematicians] are converts. Many of them didn't want to have anything to do with computers and considered that type of mathematical exploration to be inferior mathematics but, I think, the power of the computer, especially the computer graphics, has now forced some of those people to change their minds.(RW-INT-D-05b)

Traditions change slowly and, while experimental activity can suggest productive paths for research, it is still formal work, with its emphasis on proof, that legitimatizes new developments in the subject. In this debate Randy takes the side of the formalists, but he envisions a productive union between computers, experimentation and formal mathematics.

I think even many people in the field that are now using the computer as a tool know that's not enough.... Many of those, maybe even all, I can't speak for those people right now, know that ultimately to validate their work, their discoveries and computer experiments have to be expressed algebraically to fit into existing mathematics and that proof is still a necessity ultimately. But it's driving mathematics. Mathematics is now trying very hard to keep up with new discoveries made by computer. So, you know, they compliment each other in the end. It seemed like there was perhaps conflict at the beginning but it's a very happy marriage right now, I think. (RW-INT-D-05b)

Randy has personally internalized the mathematics community's continuing tension and sometimes hostile conflict between open-ended exploration and formal methods. In his teaching and visions of mathematics we will see an ongoing struggle between these two dimensions.

The alternating features of Randy's vision of mathematics show up in his first repertory grid for school subjects (RW-INT-D-01a, see Appendix J). Here physics and chemistry, the two representatives for science selected by Randy, are the subjects that come closest to mathematics, but the match is weak. The construct label, "experimental", that he provides for science also applies to mathematics, but not as strongly. For the descriptor, "investigative", Randy places mathematics midway between the sciences and history, English and French. When, for a second subjects repertory grid, mathematics is replaced by three sub-disciplines: calculus, algebra and fractal geometry; a different picture emerges (RW-INT-D-07, see Appendix K). Fractal geometry is closely matched with physics, Randy's example for science, while calculus and algebra retain the separation from science shown by mathematics on the original grid. Fractal geometry and physics are described with, "allows for expression and exploration", while calculus and algebra are "rigid". Thus, for Randy, some branches of mathematics, particularly those that are new, are experimental in nature, but some areas of the discipline retain their formal traditions.

Randy is personally excited by mathematical experiments and sees their classroom use as contributing to his goals, "for students to have understanding and procedures that are general, that is, apply in many situations" (RW-DIS-N-10).

I think the whole pursuit of mathematics can be much more enjoyable than it is and I'm trying to find some ways to make it so in my regular courses.... What I have a mind to do is to try and create a mathematics course where it's somewhat student driven, where we would do much experimentation, much less formal mathematics in, in the usual sense of the proofs and applications that require a lot of algebra and the learning of techniques. (RW-INT-D-05b)

But Randy goes on to somewhat qualify this aim by attaching it to certain mathematical domains. "The new fractal geometry and chaos topics offer us just virtually unlimited access to those kinds of activities" (RW-INT-D-05b).

 

Iteration of Functions and Pixel Rainbows

 

After the initial classroom visit that I have described above, Randy and I made plans for the research project to begin the next school year. As the Winter-Spring term wound down and we set in place a schedule for the next Fall's semester, the North City School Board announced the closing of Golden District Secondary School and transferred Randy to Northern High School. Despite losing the opportunity to teach his special course, "Fractal Geometry and Chaos Theory", Randy was determined to find a way to fit some of his favourite topics and investigations into the courses he had been assigned. He started with the obvious choice, his 12E course, a Grade 12 program that had been designated as "enriched". Here, early in the school year Randy expanded the regular curriculum to include experiments exploring the iteration of functions. Reflecting back on his planning, Randy provides his rationale for this intellectual excursion.

I wanted to present the students with some modern mathematics. I thought that there is a lot of stuff in the area of dynamical systems that's fairly easy to do. It's really very interesting, has a lot of tie-ins, especially visual ones with concepts of fractal geometry and chaos. And which really could also be thought of as an extension of one of the topics we do in the Grade 12 course anyway in composition of functions. Often the composition of functions doesn't seem to have any purpose and we certainly don't do any applications.... So I knew it was important and I also knew it was easy. We had graphing calculators and a computer now available in the classroom so we could really actually have an entire unit on it, time permitting. We started off with something simple, iteration of the squaring function - start with any number and keep pressing the square function until perhaps something interesting might show up on the calculator display. (RW-INT-D-03)

It's Monday morning and the students are involved in animated chatter concerning the weekend's social events. Mr. Walker brings the class to order by writing the lesson title on the board,

Iteration of Functions

(Dynamical Systems)

"On Friday we finished with a question that I asked you to think about over the weekend". Mr. Walker repeats the question, "Can a function be composed with itself?", and writes it on the board. "What do you think?" A number of students contribute opinions and the general consensus appears to be that the answer is yes. Mr. Walker has some sample functions for the class to consider, to ensure that all understand the concept and are ready to move on. The functions, f(x)=2x-3 and g(x)=x2-1, are written on the board and the students supply expressions for ff(x), gg(x), fff(x), and ggg(x). The the powers of x involved are noted and it is observed that, while repeated composition of f always produces a linear function, the results for g become increasingly complex. Writing on the blackboard, Mr. Walker summarizes the work so far with a note which the pupils copy into their books.

Repeated composition of a function with itself is called iteration (from the Latin "iter" meaning "journey"). Because the algebra of repeated composition (iteration) quickly becomes very messy for non-linear functions, iteration is usually done numerically, often with a computer.

The stage is now set for an experiment which Mr. Walker introduces with the title,

Investigation A: Pixel Rainbows

"We are going to use the function, f of x equals x-squared and repeat the composition over and over for different x-values." Using mathematical notation, Mr. Walker repeats this information on the blackboard.

 f(x)=x2 find fffff......(x)

 for x = 5, 1.2, .2, .8, 1.8

The pupils are instructed to get out their calculators and to think about how the iterations could be done easily. After a brief pause one student suggests, "Just put in the starting number and press the x-squared key over and over." This algorithm satisfies Mr. Walker, and responding with, "Okay, try that with each of the x's and record what happens.", he sets the class to work. The students begin these simple experiments and share their answers with each other, noting that eventually the calculators give either an error message or 0.

Mr. Walker: "What does the error mean?"

Student: "The number is getting too large"

Mr. Walker: "So what might we say the numbers are heading towards?"

Students collectively: "Infinity"

With the potential iteration results of 0 and infinity noted, Mr. Walker expands the task slightly. "Good, now I want you to repeat the experiments but this time to count how many steps it takes before the error or zero." With various students reporting their counts the following information is recorded on the board.

x = 5, 8 steps to error ()

x = 1.2, 12 steps to

x = .2, 8 steps to 0

x = .8, 10 steps to 0

x = 1.8, 9 steps to

Mr. Walker announces, "Now we are ready for the pixel rainbow part of this example.", and draws a chart on the blackboard.

0

.2

.4

.6

.8

1.0

1.2

1.4

1.6

1.8

.....

5

"Suppose we colour each number or block to show how fast it goes to zero or infinity. For example let's use red for numbers that go quickly." Mr. Walker shades in the squares for .2 and 5 with red chalk. "The colour choice does not really matter. So what colour do you want for 1.2 which took 12 steps?" One student suggests green and Mr. Walker shades in the square. Blue is given as a colour for 10 steps and yellow is assigned to 9. Mr. Walker shades in the squares for .8 and 1.8 with the appropriate colours. "There are a few more pixels to colour. Test these numbers and find out what colours we should use." The students perform the iteration on the numbers not yet shaded and report colours for .4, .6, 1.4 and 1.6.

Mr. Walker: "What happens at 0 and 1.0?"

Student: "Nothing happens. They don't change."

Mr. Walker: "Right! These are called fixed points. Let's leave them black."

Pointing to a large picture of colourful fractal images that is posted on the back bulletin board, Mr. Walker continues, "You have all seen these pictures before. The computer that drew those pictures worked in the same way we have here. It used a special function, tested each pixel or point on the screen, and then picked a colour from a table provided in the program. In the next few days we will look at this more closely."

The investigation of f(x)=x2 continues with Mr. Walker identifying examples of orbits and attracting, repelling and Julia points. With the work just completed as a model, Mr. Walker asks the class to analyse the behaviour of f(x)=/x and for homework to investigate the iteration of three, more complicated quadratic functions of the form f(x)=kx(1-x).(RW-LES-N-ii, RW-DIS-N-03a, RW-DIS-D-03b, RW-DIS-D-03c)

The investigations of the iteration of functions continue over the next two days with the students using graphing calculators and the single classroom computer to support their experiments. The fast pace of the lessons and the punch in his voice show Mr. Walker's enthusiasm for this topic. Looking back he reports his satisfaction with the outcomes of this sequence of mathematical experiments.

Very harmlessly we encountered notions of fixed points attracting and repelling fixed points, bases of attraction, neutral points or Julia points, things like that where we can develop a vocabulary. The students often find vocabulary a stumbling block, so early on in a very simple non-threatening situation we start applying names to all these things that students can see very, very easily.... And because it was all done visually I have to say that the students were engaged. Going by the number of enquiries about things they were seeing, and the ease with which I got a list of student responses to any question about what was happening I could tell they were right into it, so to speak. So that went very well and I was pleased to take them that far. (RW-INT-D-03)

 

Conflict and Retreat

 

Despite the mathematics curriculum and teaching proposals published by the NCTM (1989) and OAME/OMCA (1993) and the support provided in the introductory pages of Ontario's Curriculum Guideline (Ontario Ministry of Education, 1985), students experimentally investigating significant mathematical questions is not a common occurrence, provincially or internationally (Ontario Ministry of Education, 1991c, 1991d; Robitaille, Taylor & Orpwood, 1996). Years of teacher-centred mathematics instruction have lulled senior students to sleep and the vast majority "are content and comfortable assuming a passive role in the mathematics classroom" (Colgan & Harrison, 1997, p.7). Not long after the Grade 12E class' brief digression from the regular curriculum to explore the iteration of functions, Randy discovered that not all students and parents share his enthusiasm for modern mathematical topics.

When I phone Randy to arrange my next classroom observation session he expresses his frustrations with the Grade 12 Enriched course. Some students and parents have been objecting that the marks in his course are low. Presently, on tests, these pupils are earning grades in the 70s, while in past years their marks had been in the 90s. Randy protests, "Yes we have been looking at interesting but difficult questions. But, if a course is supposed to be enriched, challenging to pupils and providing extra opportunities for problem solving, how can the class mean be expected to be 90 plus?" (RW-DIS-N-01).

A few days later when I visit the school, Randy fills in the conflict's details.

I thought things were going well but apparently there were some students who were feeling a little uneasy, and that showed up. And I got called aside by the Department Head of Mathematics who said that, the guidance people were finding that there were three people in my class who were thinking of dropping the course because they're having a little bit of a hard time with the new stuff and because they knew that their friends in the regular math course were ahead of us. So they were starting to think now that maybe there was something negative about this enrichment course they'd gotten themselves into. And, so I found out a little bit about the politics of introducing new curriculum. (RW-INT-D-03)

The pupils in Randy's 12E class are generally from upper middle-class homes and have families that are very success oriented. They and their parents have future career plans that involve specialized university courses. Entrance requirements are high, so marks rather than real learning are the primary concern. School policies have been set to address these realities, with essentially nothing extra expected of students in enriched courses. (RW-DIS-N-01, RW-DIS-N-03a).

Students are very, very mark conscious. They're aware of what everybody else is doing, and they're pretty quick to feel uneasy when put in a new setting sometimes, although I wouldn't say that was generally true, but it did happen. And then I was reminded that the students in the enriched course in this school had, up until my coming, always done exactly the same curriculum as the others, but they did a few extra questions here and there. (RW-INT-D-03)

Randy has been going against past practices and official school policies, expanding the curriculum, and more significantly, expecting the students to learn this extra material.

It sort of was imposed upon me that I wasn't to test these students any differently than the others, which meant that the unit I had just completed on fractal geometry and chaos couldn't really be assigned a mark as if it were a test. Whereas to me it was simply covered under the topic of composition of functions and it was my way of doing that with them that I thought would be beneficial to them. (RW-INT-D-03)

As I spend more time at Northern High School it becomes increasingly obvious that Randy's teaching approach differs from that of other teachers in his department. At one point the Department Head tells me that there is no research to support the proposals made by the OAME and NCTM, and any studies that purport to favour student investigations, group work, and mathematical discussion must be flawed. All present in the department office agree. They all know that group work and investigations are a waste of time (RW-DIS-N-22).

Randy feels his isolation, but he does not attribute blame. The dominant practices in mathematics instruction have a long tradition and the system just rolls on.

The idea of throwing problem solving into a course seems to muddy the course a little bit. Some students really take that as almost extra. They feel there are ways to pass without solving problems - doing something they don't like, it's different. Um, and that impression, I think is, is the fault of the system because we try to focus on mathematical content and we never really get to the essence of the subject. Our courses always demand of us to cover so much material and we do it in the most efficient way, which is just disseminating the information. (RW-INT-D-05b)

Given the almost universal rejection of change by students, parents and teaching colleagues it is not surprising that Randy can also be observed to use the dominant transmissive mode of instruction. This is especially so when the content is traditional, such as in a 12E lesson that took place three weeks after the troubles over student grades.

The class has been studying trigonometry and for homework, using the values generated by their calculators, they were to draw graphs of the six trig functions. After working his way around the room, checking each student's work, Mr. Walker returns to the front of the class, turns on the overhead projector and displays a graph of one of the trigonometric functions. "What graph would that be?"

Student: "Sine"

Mr. Walker: "That would be sine. You can tell that it's sine by the way that it varies. It goes through the origin. It hits a maximum of 1 at 90 degrees, comes back down to 0 at 180, hits a minimum value of -1 at 270, then back up to 0 at 360, and then after that? -"

Student: "It does the same thing."

Mr. Walker, picking up on the student's answer, continues. "Does the same thing - continues in the same way. All right, that is in fact the sine."

"Now this one you would also recognize." Mr. Walker switches overhead slides to now show the cosine curve. "The point I'm getting at here is this - you must have noticed that the page with the sine and cosecant didn't look very much different from the page with the cosine and secant. If you had looked at the tables that generated these graphs, you would find that you've got all the same numbers in both tables, except that they just don't occur in the same place. I'm going to put my sine graph and cosine graph on the same axes." Mr. Walker places one overhead acetate on top of the other and lines up the two sets of axes.

Mr. Walker: "Now we are going to start shifting the one on top, the cosine, over to the right a little bit. - Now it disappears. - What does that mean?"

Student: "Horizontal shift"

Mr. Walker: "Horizontal shift - so the two graphs are identical in the sense that a horizontal shift - how much? -"

Student: "Ninety degrees"

Mr. Walker: "Ninety degrees - so we shift the cosine graph 90 degrees to the right and it becomes the sine curve. And you know from the numbers we've had that occurs all the way along."

Mr. Walker: "Any questions about either of these two curves?"

After a pause in which the class raises no questions, Mr. Walker continues. "You see these are curves that you have to get awfully familiar with. - We have to keep track of these properties that we've discovered and get evermore familiar with them. We will study lots of details in these curves."

Writing the title, "Graphs of Periodic Functions" on the board, Mr. Walker introduces the next portion of his lesson. "I'd like you to list with me some observations."

The lesson continues with Mr. Walker providing a series of short notes summarizing the horizontal shift information noted earlier and continuing with seven other facts concerning the graphs of trigonometric functions (RW-LES-N-01, RW-LES-T-01).

In the face of critical comments, from students, parents, school administration, and fellow department members, mathematical experiments have disappeared from Randy's classroom practice. Still, as we shall see, the teacher-centred direct instruction of this trigonometry lesson is not incompatible with some aspects of Randy's image of mathematics.

 

Mathematics as a Technical Language

 

The previous lesson's emphasis on precisely noting the properties of trigonometric curves and carefully recording these observations in correct mathematical words and symbols captures Randy's view that "math is like a language" and students "need to develop the ability to converse and follow an argument" (RW-DIS-N-07). This theme of mathematics as a formal language with precise rules arises often in Randy's writing, repertory grids, and interviews.

In justifying the existence of mathematics as a secondary school subject Randy writes, "Mathematics is the language of many technical subjects, the language of problem-solving" (RW-DOC-D-07). When developing his first school subjects repertory grid, Randy provided the pair of opposite descriptors, "languages" and "science", to separate the disciplines. Although on most constructs mathematics is closest to physics and chemistry, along the "languages-science" axis Randy places mathematics nearer English and French (RW-INT-D-01a, see Appendix J). While English, French and mathematics are "languages" and "mental" activities, there is a distinction; mathematics is "quantitative" while the two more conversational languages are "qualitative" (RW-INT-D-01a). All three languages: mathematics, English and French; are described as "traditional" (RW-INT-D-01a), that is, they have a history to respect.

When, on a second school subjects repertory grid, mathematics was divided into separate strands, Randy described the traditional school content of algebra and calculus as being a "technical language" and "concise" "symbolic expression" (RW-INT-D-07, see Appendix K). Although not as strongly, these labels are also attached to the newer mathematical strand of fractal geometry. Randy sees a need to help students develop a precise language in this new sub-discipline, for "some of the vocabulary of dynamical systems will be encountered by many students in their future careers" (RW-INT-D-03).

As his opening sentence in response to the question, "What is mathematics?", Randy writes, "Mathematics is a body of knowledge, a set of rules, tools and techniques" (RW-DOC-D-07). Mathematics, this collection of rules and techniques, deals "with things" rather than "with people" (RW-INT-D-01a) and is "objective" and "rigid" (RW-INT-D-07). "It's not just all fun and games and say unstructured play, but it's a discipline that has certain measurable aspects that you can count on" (RW-INT-D-03).

Reflecting his view of mathematics as a structured discipline, Randy's lessons are carefully planned and organized. During an interview I comment on Randy's extensive lesson preparation and structured approach and ask if he sees this as a quality of good teaching. Randy's response is mixed.

I think it's somewhat ambivalent. I don't think it's necessarily good or bad, although it tends to the good.... The danger is you don't let any students get in your way of completing it exactly as planned. So it's not necessarily a quality all the time.

This mixed view is characteristic of both Randy's conception of mathematics and teaching practice. Apparently opposing components of his subject image, such as the discipline's growth through open ended experiments and the view of mathematics as a rigid formal language, are reflected in instructional methods that appear to come from opposite ends of the spectrum.

 

Mixed Practices

 

Randy's image of mathematics as a precise formal language comes through most clearly in his teaching when he is introducing new content from the traditional core curriculum. Here his lessons follow the North American tendency to emphasize definitions and vocabulary (Center for Science, Mathematics, and Engineering Education, 1996).

As the previously introduced trigonometric graphs lesson continues and Mr. Walker provides concise blackboard notes for the students to copy, he turns to the topic of "Reciprocal Pairs".

Mr. Walker: "When sin equals 0 what about csc? -"

Student: "Undefined"

Mr. Walker: "Undefined - There's a gap there. What we call a discontinuity. It's discontinuous. And where does this occur?"

Student: "Zero, 180, and 360."

Mr. Walker: "Zero degrees, 180, 360 - whenever the sine graph is 0."

The first line of the next note is put on the blackboard, "when sin=0, csc=undefined 'discontinuity' (at 0, 180, ....)", and Mr. Walker continues. "What does the graph look like near where it would be undefined. I can't ask you what it looks like here (pointing to 0 on the projected graph), because it doesn't look like anything. Right? - It doesn't have a point at 0. What does it look like near there?"

Student: "It is going to go straight up. - It follows an asymptote."

Mr. Walker: "Okay - so csc approaches - what kind of asymptote? -"

Student: "Vertical."

Mr. Walker completes the note with the second line, "csc approaches vertical asymptotes when sin=0."

Mr. Walker: "You've got to be really careful in mathematics. We are so picky about the way we use words" (RW-LES-N-01, RW-LES-T-01).

Although almost the full 75 minutes of this lesson featured such precise examination of the trigonometric graphs, there was an aside at about the mid-point in the period that both underscored Randy's focus on mathematical language and illustrated his potential to shift direction and engage students in rather open ended explorations.

It is coming up to the middle of the semester and the date when mid-term grades are released. At a pause in the lesson Mr. Walker takes time to outline the process for grade calculation and to comment upon and return recent work that he has marked.

Mr. Walker: "I was very pleased with the notebooks that I marked over the weekend. Most of them are going to end up with a mark of ten out of ten. Almost all the others are very close to that. There were just one or two that were low. - To keep a notebook all you have to do is to copy the notes off the board, have them dated, titled. I am going to start asking though that you underline the titles. A few of you are very economical with your paper, and one note starts right in the middle of another. There is no spacing so underlining will make them a lot easier to use, for yourself especially. I only get to see them occasionally but you get to see them everyday."

After a short pause to let this message be absorbed, the announcements about assignments and grades are continued. "Notebook mark was good. Notebook counts out of ten. - And then I thought that a really very good effort was made on bonus assignments. I know that it took me forever to mark them. That's just because I was reading them all the way through. There was really a lot of good stuff there. Not only are most of you doing a good job on it, but some of you were doing unbelievably good work in exploring the question even more deeply than I asked you to, and coming up with some new mathematics. I'm impressed and the marks will reflect this effort" (RW-LES-N-01, RW-LES-T-01).

As the file of bonus assignments is passed around the class, the students retrieve their work and the focus of the lesson returns to trigonometric graphs. I ask Stephen, the student who sits in the desk beside me, if I can take a look at his work. He searches through his pile of assignments and quickly selects an example with the title Circles Squared and the "Superegg". His eagerness to share and his smile suggests that he is rather proud of his work. A quick read of the assignment shows that this pride is fully justified.

Stephen's introductory paragraph provides a good summary to his project.

I noticed that the equation of a circle, y=±/(r2+x2), can be slightly altered to form almost a dozen different shapes. If we use y=±e/(re-xe), and give e different values, 10 distinct shapes can be found.

The rest of the opening page contains ten sketches of graphs that resulted when e was assigned a single value such as 1, .5 or -.5, or more generally came from subsets of the real numbers such as "e = odd integers and e>1". Stephen's work goes on to conduct the same explorations using the equation for an ellipse as the starting point (RW-DOC-D-04).

In an interview, conducted a few days after the trigonometry graphs lesson, Randy talked about the assignment of bonus questions and his experience with them so far this term.

There is usually a spontaneous nature to the whole affair. Sometimes, something related to the current work comes up that seems a little bit more interesting or a little bit too deep to pursue in the class within the regular schedule, so I try to devise some way of getting the students to explore it on their own. Usually there'll be a bit of an introduction to the question given in class, then they're assigned to try to follow that up. Other times the question arises spontaneously from a student question or, or a student discovery. There's a few of the students in that course who will actually go home and come up with some fairly creative questions. (RW-INT-D-03)

School policies, having denied him the opportunity to take class time for mathematical explorations that extend the regular curriculum, Randy has turned to a strategy of bonus questions and marks. Students are expected to submit work, but there is little risk, as the marks received can not reduce their regular course grade.

They're obligated to hand in something. That's the commitment they make when they sign up for the course as far as I'm concerned. And some of the responses you'd have to describe as feeble. They're going through the motions. There actually turns out to be a few students that don't seem to have been interested in any of the questions. But from the amount of time they've spent on them, it looks like they wouldn't know enough about the question to know if it's interesting. They just don't feel like doing it. And a number of them will do fairly well on anything quite mechanical, but only a few of them are risk takers enough to actually give responses that you can call original, and those are usually much better rewarded. (RW-INT-D-03)

Randy appears ready to accept the fact that some students will not respond to his invitations to mathematical adventure and he is happy to focus on the successes. The interview continues with his description of these.

There's often an investigative aspect to it and, again much of the time it comes spontaneously from the students. They just see more in the question than I expected and I've been getting some stuff that is just absolutely excellent, that I end up copying so that the next time I ask that question I'll realize just how much it involves. (RW-INT-D-03)

Randy brings out a copy of a sample assignment, Stephen's work that I examined earlier in the week.

As far as I can remember, I gave him no kind of prompting at all for that question. That was completely his own.... I was very intrigued by it, especially by the level with which he pursued it and the different dimensions of it that he looked at. (RW-INT-D-03)

On issues of mathematics pedagogy Randy appears to be of two minds. When addressing the traditional curriculum, the formal mathematical language, he is at home using a teacher-centred approach. His lessons are well prepared and through the use of direct instruction he manages to execute his plans. On the other hand, when extending the curriculum or introducing new non-traditional topics, Randy's preferred approach is the use of rather open-ended independent student investigations. In many ways Randy's mixture of teaching practices and somewhat fractured discipline image parallels the debates presently taking place within the mathematics teaching profession.

 

A Personal Philosophy Under Construction

 

Randy's conception of the nature of mathematics appears to be in transition and does not fit conveniently under any of the labels provided in schemes for categorizing philosophies of mathematics (Carnap, Heyting & Von Neumann, 1983; Davis & Hersh, 1981; Ernest, 1991; MacLane, 1986). His struggles with some of the tasks involved in this study and comments made during the subsequent interviews show that Randy is in the process of reformulating his personal philosophy of mathematics. Recent study and reading have raised new questions for Randy, but on a number of these issues he indicates that he is not yet ready to take a stand. "Well there are a lot of ideas in philosophy that I don't find I have to take sides with. - It's just interesting to listen to the whole thing. They're probably unanswerable anyway" (RW-INT-D-05).

As Randy works on his school-subjects repertory grid, searching for descriptive labels for the disciplines, he reports, "I can not believe how hard I find this" (RW-INT-N-01). He often goes back and revisits the indexing he gave on past constructs, making minor adjustments in the rankings provided for subjects. At times Randy becomes frustrated and while working on one particularly difficult construct asks, "How long should I puzzle at this at any stage? I change my mind back and forth as I think about each description" (RW-INT-N-01).

Even after the grid is complete, Randy is not sure of his description for mathematics. Reflecting on his choice of "deals with things" as opposed to "deals with people" as a feature of mathematics, he rethinks his position and suggests that it could change.

I'm not sure what I'll think tomorrow about dealing with people and dealing with things. Maybe there is a mark of a human aspect in mathematics as we're solving people's problems. We may, you know, get into a social issue or whatever, but that's an idea I really have to develop a little more before I want to say much about it. (RW-INT-D-01b)

In the end, after considerable careful thought, Randy is able to attach definite labels to most of the disciplines and the result is a dichotomous arrangement on the PrinCom display (RW-INT-D-01a, see Appendix J), with two separate clusters of subjects: the sciences on one side and the arts, languages and humanities on the opposite. Mathematics is more difficult to categorize and on six of the ten constructs produced, Randy places mathematics in the grey, middle ground between the opposing descriptors. Looking at the display, Randy acknowledges his mixed view of the subject.

Well, it looks like a group - the arts and sciences at opposite poles - and mathematics, although it's on the side of the sciences, it certainly seems to be off by itself.... It's harder to classify is what it seems to be saying. It's harder to say where mathematics exactly fits in. It's kind of between things, rather than being easily pinned down. (RW-INT-D-01b)

In particular, Randy places mathematics in the middle of the "truth determinable-truth indeterminable" scale and thus does not commit to either the absolutist or fallibilist position (Ernest, 1991). His recent studies in fractal geometry and chaos theory have shown Randy that some branches of mathematics are "much nearer to rapidly evolving and experimental and investigative" (RW-INT-D-01b) than traditional school mathematics, but this does not really alter his overall image of the discipline. Moving mathematics on the PrinCom display and thus, "putting it closer to some of those other things though really wouldn't be consistent with what I'm thinking. You know what I mean, truth indeterminable, I don't think it's any closer to that, or belongs any closer to that" (RW-INT-D-01b).

Randy is not ready to take a side in the debate between Platonism and constructivism and responds to my questions concerning the sources of mathematical knowledge with:

The discovering versus the inventing of mathematics - well I don't know where I stand on that. Um, it's probably a little bit of both. I think some days I could give you a better answer than other days. But I know it's an interesting debate among the experts themselves as to whether or not mathematics is invented or discovered....If the universe is put together mathematically then certainly we're discovering some of its workings and that becomes part of mathematics. But, it seems like the universe is not any longer a giant clock, at least in a dynamical system, and that affects all the basics that we seem to understand about the universe and mathematics. (RW-INT-D-05b)

The outer ring of Randy's concept map for mathematics (RW-INT-D-09a, see Appendix L), with the label, "Creators of Mathematics", appears to suggest a constructivist position, a view of mathematics as developing out of human thought. But, this is not completely Randy's intent, for he reserves the title "creator" for only a few key mathematicians.

What I had in mind as I was writing that was certain very prominent people in the history of mathematics who made enormous individual contributions to the subject ....It seems to me that, although there are many mathematicians making contributions, the contributions of some were just huge. They were almost super human in their abilities to grasp mathematics, to relate it to solve difficult problems quickly. (RW-INT-D-09c)

It appears that Randy does not see mathematics as part of general human culture, something to which all contribute.

Without a strong social constructivist conception of mathematics, the epistemology that underlies the NCTM reform proposals (McLeod, Stake, Schappelle, Mellissinos & Gierl, 1996), Randy is not ready to fully adopt teaching practices as outlined in the Standards (1989). These he reserves for new, non-traditional topics. "When studying fractals it's largely student activity. The students sit down with a question basically, and then they do some exploring, gathering data or somehow investigating the problem" (RW-INT-D-06). Given the constraints provided by the present school system, Randy argues for a less adventurous approach for the core curriculum.

I'm sure other topics could be taught in the same way, but teachers have a lot of time pressure on them. In order to build a course like that you have to sit down and think about it with lots of time to do it. And another thing is that, you don't cover as much content in as much detail, and you don't master skills such as you have to do in algebra to the same degree. What you may do is, I think, make an awful lot more connections in math. And the concepts may become clearer but you may be sacrificing certain skills that some people might want to see in the course. So somehow you find a balance. (RW-INT-D-06)

 

Mathematics as Problem Solving

 

Randy believes that his subject is an important component of the secondary school curriculum, since "mathematics not only enriches one's general education, but continued study in the program will keep many career doors open" (RW-DOC-D-07).

I think the that the reason for mathematics being a prominent subject in education is the number of ways that it can be applied. If you ask the average parent for one of the most important subjects that the students take in high school, I'm sure that mathematics would be your number one or number two and I think that's because of the perception that almost every subject requires mathematics, requires you to use mathematical thinking or mathematical language or mathematical problem solving. (RW-INT-D-09c)

Concern for his pupils' future employability leads Randy to personally explore applications of mathematics and look for examples to bring to the classroom. But, his reasons for interest in the uses of mathematics extend beyond student needs. For Randy, the discipline and its applications are intertwined. "I can't really justify the existence of mathematics all by itself. There has to be a reason why the system exists" (RW-INT-D-14d). As Randy will explain in the following paragraphs, mathematics and its applications can not really be separated.

In writing about the nature of mathematics, Randy states that the discipline "has developed, originally, out of necessity in areas such as land surveying, navigation, commerce and war" (RW-DOC-D-07). This theme of mathematics arising out of human efforts to solve practical problems is further developed in Randy's concept map for the subject (RW-INT-D-09a, see Appendix L). Here he shows a ring labelled "APPLICATIONS" drawn around the cluster of nodes that represent various branches of mathematics. Arrows run from a "Creators of Mathematics" (mathematicians) node through this ring into the body of the discipline. Randy explains the meaning of this picture.

Applications along the periphery are not meant to indicate any kind of secondary importance it's just a way of putting them together. I tried to indicate that I think in the history of mathematics problems, real needs, led to the mathematics. (RW-INT-D-09c)

Nineteen of the thirty-six nodes in the concept map represent activities, such as medicine, music, art, psychology and sociology, that others might not view as mathematics. Randy asserts that these belong in a picture of the discipline since they represent "real world problems of some kind that are solved using mathematical techniques or mathematical knowledge" (RW-INT-D-09c).

Along with applications, Randy's concept map shows another motivating force for the development of mathematics, "Desire for Understanding (Philosophy)". This human drive has led to "much modern mathematics that derives from pure research interests" and generally "runs ahead of applications development" (RW-DOC-D-07). Randy sees a synergistic relationship between pure mathematics and other disciplines.

On the other side of the coin certainly there are pure mathematicians who may not have any interest in solving any practical problems but still produce mathematics that someone else sees in a different light. There are some fairly well-known instances of physicists who have gone to some pure mathematical concept to fill in some blank in the structure of matter that they were exploring and in some cases had even such confidence in the mathematics that predictions were made as to say things such as the existence of particles that had never been detected before. And whole experiments sometimes, expensive I assume, were constructed just on hypotheses that were based on purely mathematical considerations....So there are instances of course, of the mathematics preceding the application. I think it works both ways. I think it's very much a human endeavour and there's no straight-line path from problem to solution. (RW-INT-D-05b)

The significance of applications in the history of mathematics implies, for Randy, an importance in the teaching and learning of the subject. "As I view my job as teacher today the applications have to be there. The applications are the reasons for the mathematics....I don't think we could justify the existence of the mathematical infrastructure in the education system without that" (RW-INT-D-09c). Unfortunately mathematics education, according to Randy, fails in this task. "It is somewhat doubtful, in my view, that university and probably even high school mathematics finds many direct applications in the lives of most people, beyond simple arithmetic. As of now we have not determined how to deliver useful, meaningful, mathematics to the majority of our students" (RW-DOC-D-07). In a subsequent interview Randy expands on this idea.

I realize that when I'm expressing that thought, I'm revealing some of my own ignorance about how some of the current mathematics might actually have more application than I'm aware of. Notwithstanding that, I've come to believe that a large amount of it doesn't have any relevance for most of the students and maybe some of it has no relevance for any of the students. That we may not even know why we're still teaching it except that we always have....I think that our mathematics courses seem to be designed to make students into images of math teachers or people who have gone before and we haven't put enough thought into what valuable skills or topics we could include in our mathematics courses. (RW-INT-D-05b)

Randy realizes that in the past he has been part of the problem, but he has recently begun looking more closely at applications.

I was always the teacher who delighted in the mathematical ideas and knew there were applications out there somewhere, but the students would meet these in an appropriate time when they began to specialize and choose their careers. And so I felt it was my job just to make the ideas as clear and interesting as possible. But the applications didn't seem to be very many at the high school level nor was I particularly inclined towards that kind of thing....I am beginning to realize the problems people do in physics or in industry are really important and they're quite interesting areas. Before I'd say, I hadn't paid much attention to them and didn't consider them to be as interesting as the pure mathematical ideas. (RW-INT-D-09c)

Randy also objects to the common textbook approach, where any application is "pared down until it comes out looking like all the other questions" at the end of the chapter (RW-INT-D-09c).

I don't think it works....You may get there but I don't think that works well at all. I don't think that's the way people learn....They can learn mathematical algorithms but the understanding won't come about until they use it and can see, hands-on in some way, what it is they're doing.... Mathematics is the abstract, you don't start with the abstract. By the time the students and the teacher get to the end of that mathematics topic, at least their interest is very much exhausted so that the applications often just get paid lip service. (RW-INT-D-14d)

In his recent studies, Randy has found that he personally learns best when he starts with a problem and moves to the mathematics.

I really get excited about mathematics when I have a problem in front of me that I don't know how to solve and I start to explore and I occasionally look up something that I don't know or don't remember. (RW-INT-D-09c)

He sees this problems-to-mathematics order as best for his students. He explains using the topic of graphing as an example.

Start from the concrete and move towards the abstract, so the science should come first and the learning of the graphing done with the data and with the motivation to try to come to an understanding of the data, and then the mathematician might refine that. Because after all, I think that's what mathematics is, it's just the refinement of some of the abstract ideas, taken out of applications. It may be the history of the subject but it also makes a lot of sense from the point of view in the way people learn. (RW-INT-D-14d)

But, as Randy has found with other innovative teaching ideas, "to use stuff like that you really need lots more time than you have...You've got to work it within your time constraints and those things take up much more time than I think most people would imagine" (RW-INT-D-09c). As we shall see, the constraints of the system mean that sometimes Randy has trouble putting his plans into action.

 

Connecting Science and Mathematics

 

Along with his mathematics classes Randy also teachers a Grade 9 science course, part of Northern High School's special program linking science and technology. Over the past three years the school has been experimenting with the curriculum for this program and as a result there is no fixed prepared outline for Randy's course. He is frustrated with this "seat of the pants planning", for it conflicts with his desire for organization and careful preparation of lessons. Still, the lack of a course outline has some benefits, and Randy uses this as a justification for taking his class off in directions that he wishes to explore (RW-DIS-N-02).

During a unit on the particle model of matter, Mr. Walker provides a demonstration, dropping a piece of potassium permanganate into a beaker of water. The solid disappears and a purple colour spreads out through the liquid. Why? The class postulates that the potassium permanganate particles must separate from each other and become suspended in the spaces between the water particles. But why does the colour move through the water solvent? The pupils suggest that the potassium permanganate particles are bumped by the water particles and are thus pushed along. Mr. Walker, pretending to point to one individual solute particle in the middle of the beaker, inquires, "On what side will the bumps occur?" "All sides. From all directions, equally", is the popular student opinion. Mr. Walker continues, "So, why does the particle move? Do you think the particle should move or stay in one place?" After a brief debate the class holds a vote. Despite the evidence provided by the spreading purple colour, the majority believe that the potassium permanganate particles should stay in one place. Mr. Walker takes this science puzzle as an opportunity to develop some mathematical ideas, random events and probability, two concepts that he has observed are problematic for many people.

There are difficult concepts there. There are things that you sort of don't want to touch, might not be able to answer, or complete usually. So I think they've kind of just been filtered out of the curriculum whereas I consider them to be basic....We take it [randomness] for granted and there are things to know about random processes. So, I think that's something important that I could address, something that university students don't have a good grasp of at all. Students flounder in probability courses originally because they don't have any simple mental constructs to deal with some of the concepts. I think that the students learn how to do the problems, but I'm not sure they ever learn very much about probability. That's what I have in mind. That's what drives me to do it. (RW-INT-D-02)

Mr. Walker suggests that the class imagine looking at just one solute particle and focus on its progress along one line across the beaker.

Mr. Walker: "Along this one line, in how many directions can the particle move?"

Students: "Two, left and right."

Mr. Walker: "According to our theory why might the particle move?"

Students: "It gets bumped by water particles."

Mr. Walker: "How do the chances of a bump from the right compare to the chances from the left?"

After some debate, the class concludes that the chances of a collision from either side are equal or each 50 percent.

Mr. Walker: "We are going to do an experiment or simulation of this situation. We need some way of deciding if a bump will come from the right or the left. We need something that is not predictable but has only two possibilities. Any suggestions of what we could use to tell us the direction of a bump?"

A number of pupils suggest tossing a coin. Mr. Walker is ready for this, pulls a penny from his pocket and tosses it onto the lab bench. It comes up heads. "What do you want to call that, moving left or right?"

The popular response is "right", so Mr. Walker writes "Heads - moves right" on the blackboard and puts "Tails - moves left" below this. Two more tosses of the coin give another head followed by a tail. On the blackboard Mr. Walker charts the progress of the particle, showing that it is now one step to the right of its starting point. Mr. Walker continues, "Of course that is just one possible example of three bumps out of thousands of collisions. We need to look at more."

Mr. Walker describes what he wants the class to do for homework. Each student is to toss a coin 100 times, record the results, heads or tails, and graph the progress of a solute particle. To finish they must note how far the particle is away from the origin after the 100 bumps (RW-LES-N-04).

Next day the class begins with Mr. Walker asking for the results from the homework simulation activity.

Student: "You won sir!"

Mr. Walker: "How? I did not say anything about what might happen. I just asked questions."

Student: "Our theory was that it would stay in the same place. Mine moved."

Mr. Walker: "Who else found that their particle had moved by the end of 100 collisions?"

All but two of the nineteen students in the class report some total migration of the solute particle.

Mr. Walker: "We will not worry about left or right, just how far away the particle is from its starting point."

The results from the class are collected on the blackboard and an average distance of 11.1 is computed. With this new data in mind Mr. Walker encourages the class to re-examine their hypothesis. After a short debate a re-vote is taken. Again a majority vote for the solute particle staying in one place, but the margin is slightly reduced.

Mr. Walker: "What if we tossed the coin more times, a thousand times, a million times?"

As the debate continues a variety of opinions and arguments are offered, the majority of which focus on the idea that there should be equal numbers of heads and tails and thus they should cancel out.

Later, after the completion of this unit and Randy and I discuss the sequence of lessons, he reflects back on the class debates and votes.

It seems that kids have enthusiasm for that. They like the idea of talking about things that take you to the edge and I think they appreciate the idea that there are a lot of things that people can't answer. I mean, we have discussions on randomness so they get to express themselves. They get to take some chances in the discussions that they might not otherwise do when we're dealing with topics of certainty. (RW-INT-D-02)

As the lesson continues, Mr. Walker leads the class in the development of formal notes recording the coin toss experiment and the resulting data. As this is done the vocabulary and concepts of random events, Brownian motion, and random walks are introduced. Finally, returning to his question of larger samples, Mr. Walker turns to the computer that he has brought to class and runs a coin toss simulation program that reproduces the students' work for a sample of 2000 tosses. A distance of 44 steps from the origin results (RW-LES-N-08).

The initial question, "Why does the purple potassium permanganate colour spread?", originating in a science experiment, motivates two more class periods of mathematical exploration. Together, Mr. Walker and his class develop an argument as to why a solute particle could be expected to migrate away from its starting position and establish a function linking the number of bumps or coin tosses to the expected distance of travel. Freed from the constraints of a packed rigid curriculum, Mr. Walker takes the opportunity to expand the results of a simple science demonstration into a deep exploration of probability and random events.

A different picture emerges when, facing the pressures of a packed Grade 12 mathematics curriculum, Randy addresses the traditional topic of trigonometric functions. In previous pages I have described the interaction between Mr. Walker and his Grade 12E class as they looked at trigonometric curves. Later in this lesson, when introducing the words, amplitude and period, Mr. Walker made a brief digression to provide context and justify the topic.

Mr. Walker: "Does that [amplitude] sound like a word from physics? - You can produce graphs like this in physics. You could hook up a microphone to an oscilloscope. A microphone will convert sounds from your mouth to an electrical pulse. The oscilloscope then will display the pulse as a vertical displacement, but it will also change with time. In other words, time would be the horizontal axis. If you whistled into a microphone you would get a sine curve. And if you blew harder how would it affect the amplitude?"

Student: "It would go higher."

Mr. Walker: "So that's a word right out of physics. - And if you blew a higher note, what else would change?"

Student: "The period."

Mr. Walker whistles two notes, the second with a higher pitch, and pointing to the peaks of the sine curve graph continues, "You get more of these over the same amount of space, so the period is shorter. So we are getting words out of physics. There are a tremendous number of applications of periodic functions in physics and elsewhere. The mathematics of these simple curves is a frequent model for a number of physical systems. We will have a look at some of them, but first we want to get some of the mathematics down."

Here, although Randy acknowledges that the mathematics being studied can be used to model real concrete phenomena, the beginning point is the mathematical abstraction of trigonometric functions and the accompanying vocabulary details. As the trigonometry unit progressed, time pressures restricted applications to brief asides such as the above.

 

Proofs and Mathematical Truth

 

In his initial writing about the nature of mathematics, Randy expresses a strong absolutist position, stating that "mathematics, more than any other field of knowledge attempts to deal with truth....Mathematics is equally concerned with truth and with number" (RW-DOC-D-07). But, when exploring his vision of mathematics more deeply through the construction of the school subjects repertory grid, Randy reveals that his image of mathematics as truth might be less strong. Here on the PrinCom display (RW-INT-D-01a, see Appendix J) mathematics falls some distance away from the labels "consistency required" and "truth determinable" and Randy, while expressing some surprise at this arrangement, does not reject it outright.

It's not as close to consistency required. I'm wondering what pulls it away from those things that we often associate with mathematics. Truth indeterminable almost seems to be in the wrong end of that line. Truths are very determinable in mathematics if we have the proper assumptions to begin with....It seems quite strange. (RW-INT-D-01b)

When discussing the links between mathematics and the natural world, Randy further questions the discipline's claim on truth. Although "mathematics has helped humans gain some understanding and control over nature" (RW-DOC-D-07), Randy rejects "the Laplacian reductionist view" (RW-DOC-D-07) and holds that this understanding is neither complete nor absolute.

It was largely believed by philosophers, mathematicians, and scientists that we almost knew everything there was to know. There were just a few details to fill in about how nature worked. And then, of course, quantum mechanics comes along and we realize the probabilistic nature of some truths. And now in the age of dynamical systems we're learning more about what we don't know or can't know, about how things are unsolvable....That's an idea that we hadn't accepted for a long time in western culture. We thought we were on the verge of conquering nature and now we find out that we can't even understand nature, let alone be the controller of it. So it's a pretty deep and important, I think, fundamental limitation of human beings that we're learning about. Now, who would have thought that mathematics would teach us that? Mathematics was supposed to be about finding truths and answers and the knowable. (RW-INT-D-01b)

Randy is also in the process of re-thinking his position on the nature of mathematical proof. During Grade 12 lessons examining the trigonometric solution of triangles Randy rejects his students' loose arguments for the equality of lengths and angles and insists on the use of formal Euclidean geometry. He justifies this stance in a brief aside on the nature of mathematics.

There is a danger in geometry in trusting one's eye to give equal angles or lengths. We need to prove everything. That's how mathematics works. We start with things that all of us agree are true and argue from there. Anything we prove today can be used tomorrow to continue our work. We always need to refer to past proofs. (RW-LES-N-15)

In an interview, Randy further develops this view of mathematics as a formal system and links this to learning within the subject.

Of course that's how mathematics grows. That's how Euclid developed his geometry it seems, or at least that's the way he presented it to us. And that's the way we learn, period. We form concepts that help us to understand other concepts. It's the building upward sure and then building outward as well. And mathematics pays an awful lot more attention to that than other disciplines do, I guess. We try to make sure that our mathematics is consistent every which way. So we're always checking it to see that it doesn't contradict the earlier things that we assumed, that were fairly obvious to everybody. It's one of the essences in math, I'm sure. (RW-INT-D-05b)

As the interview progresses Randy softens his stance and admits some doubts.

Mathematical things are agreeable to anybody who can understand what is being said. At least, that's what I was told and came to believe for quite a long time. I'm aware now of some mathematical proofs that are so difficult that there might be only a few people who understand them. In which case, it might be challenging the notion of what a proof is and hence making it much more difficult for me to answer questions of truth. (RW-INT-D-05b)

In Randy's emerging, less formal image of mathematics there are different levels of proof and indisputability.

I think that what makes truth, truth, is indisputability at its very highest and maybe lower than that there are some graduations of convinciblity where there seems to be overwhelming evidence; be it by virtue of a large number of examples, a complete lack of counter examples, and otherwise strong evidence either in the form of examples or logical construction. But, I'm sure the new proofs in mathematics are making everybody rethink. The proof just done by Wiles, of Fermat's Last Theorem is much too difficult for a large number of people who are not in certain specialized fields to understand. So that's getting away from the original idea of proof. I may not really appreciate what mathematical proof is anymore because I used to think it was a lot simpler. (RW-INT-D-05b)

Randy's recent consideration of alternative definitions for mathematical proof interacts with his teaching experiences and his students' struggles with the more formal aspects of mathematics. Although "simplicity and clarity are the essence [of mathematics], the mathematics student will not always agree that such demonstrations are successful" (RW-DOC-D-07). Even when proofs "use only relatively elementary algebra students have troubles following the logic. They can follow the proof, but it's not very convincing. Experimental approaches are stronger. I have found that the algebra excludes a number of people" (RW-DIS-N-08b). "They know that it's right, but they still don't always feel that it's right in their system. Their intuition just isn't along the same lines as our logic. They construct their world differently and we're imposing ours on them" (RW-INT-D-05b). Randy realizes that formal logic, while mathematically correct, does not always produce a convincing argument for students. "They really have to have more than just their logic satisfied it's got to feel right too. We don't function on a logical level most of the time so that's not satisfactory" (RW-INT-D-05b). He believes that in mathematics teaching, "we just overlook the tremendous power of intuition which drives students" (RW-INT-D-05b), and in his lessons Randy employs visual proofs that appeal to his pupils' intuition.

 

Visual Proofs: Helping Students See Mathematics

 

Randy's Grade 12 class is continuing its study of trigonometric functions and there is a request to look at one of the homework questions; "Show sin(180 - h) = sin h". On the board, Mr. Walker draws a sketch of the sine function for a full cycle from 0 to 360 and turns to the class. "Any suggestions?"

Student: "Could you pick a particular angle? I tried ninety degrees and then they're equal. You get sine ninety on both sides so they are equal."

Mr. Walker: "That's true, but it does not prove that the equation is true. We need to show it for any angle. That's what we mean by a proof in mathematics."

Mr. Walker calls for a proof, but in fact, he sees a general demonstration as sufficient and proceeds to develop one using his sketch of the sine curve. "Suppose we measure out an angle of size alpha along the x-axis." Mr. Walker draws an arrow along the horizontal axis, starting at the origin and pointing right. He labels this h. "Now where is a hundred and eighty degrees on my graph?"

Student: "Where the curve crosses the axis."

Mr. Walker marks the 180 point and continues. "On the graph what would it mean to take away alpha? What move do I need to make to subtract alpha from one hundred and eighty?"

Student: "Move left."

Mr. Walker: "Yes, but how far?"

Student: "Same distance you moved before on the left for alpha."

Mr. Walker, satisfied with this answer, draws an arrow of length along the horizontal axis, left from 180, and marks the distance as -h. He points to the tip of the first arrow (h) and continues. "This is the angle alpha. How do we find the sine value here?"

Student: "Go up to the curve."

Mr. Walker responds with "Right", and draws an arrow up to the sine curve. "How would we find the sine at one hundred and eighty minus alpha?"

Student: "Measure up to the curve at the point of the other arrow."

Mr. Walker, responding with "Good", draws a second vertical arrow up to the sine curve.

Mr. Walker: "What do you notice about the lengths of these two vertical arrows?"

Student: "They're equal."

Mr. Walker draws a dashed horizontal line across the tips of the two arrows and over to the vertical axis

 Mr. Walker: "What does that tell us about the sine values then at alpha and a hundred and eighty minus alpha?"

Student: "They're equal."

Mr. Walker: "There is nothing special about our example for alpha. We could repeat all this for other alphas. So we have a proof" (RW-LES-N-11).

On this occasion, the above simple, but effective visual demonstration serves as a proof, but Randy is not always willing to be this non-rigorous. Two weeks later, when introducing the topic of trigonometric identities, Randy insists that the students go beyond graphical demonstrations of equalities.

The pupils enter the equation, y=(sin M}2+(cos M)2, into the graphing calculators that Mr. Walker has brought to class. Graphing the equation they get the horizontal line, y=1. Mr. Walker acknowledges the students' "discovery" that sin2M+cos2M=1, and writes the equation on the board. But, the graphical evidence is not sufficient and Mr. Walker insists, "We need to prove it. We can see it on the calculator and it's pretty convincing, but we need a proof." The lesson continues with Mr. Walker, using the basic definitions of sine and cosine, leading the class through a formal proof of the Pythagorean identity (RW-LES-N-19).

In the previous section, while discussing the nature of mathematical truth, Randy showed that he is in the process of developing a flexible definition of proof. Randy, as he says, "may not really appreciate what mathematical proof is anymore" (RW-INT-D-05b), but he is not willing to completely abandon formal algebraic methods in favour of visual demonstrations.

 

Mathematics as Art

 

Randy sees strong connections between mathematics and the arts, and indicates this in his concept map for the subject (RW-INT-D-09a, see Appendix L), where nodes labelled "Art" and "Music" are directly linked to another for "Geometry". When later explaining his map, Randy expands on this connection. "I just think in a person's brain, if you had studied geometry and art both at some time, then connections would be made. You know, perspective, pattern, and organization, spatial organization are all geometric attributes" (RW-INT-D-9a). In fact, for Randy, mathematics, or at least some branches of the discipline, is art.

It's obvious to anybody that there's beauty in mathematics, and fractals in particular, but in much of mathematics, especially geometry. And whether or not someone wants to define it as art, that's a matter of opinion I guess. But as the old saying goes, "I may not know art but I know what I like". Fractals are beautiful and they are visual, so we can't escape the connection to art. (RW-INT-D-08b)

Among the books on Randy's desk in the mathematics department office is a large format volume of computer generated fractal images (Peitgen & Richter, 1986). Often, for relaxation during spare moments, Randy opens this text and carefully studies a picture. Sharing classrooms with teachers of French and history means that Randy has limited bulletin board space available, but what he has is covered with commercial posters of fractal art.

The artistic dimensions of Randy's image of mathematics are further developed in his subject repertory grids (RW-INT-D-01a, RW-INT-D-07, see Appendices J and K) and in interviews where he reflects upon the grid PrinCom displays. In the first grid developed (RW-INT-D-01a), Randy's labelling of the subjects produced two clusters; the sciences, chemistry and physics, side by side on the right and music, languages and history together on the left. Mathematics sits alone, some distance from all other subjects. Randy responds positively to the message contained in this arrangement.

I think it's kind of nice to see mathematics out there a little bit by itself, because we tend to get labelled. People outside of our discipline kind of dismiss mathematics rather quickly and probably lump us in with technical subjects much more than we belong. I think mathematics is still considerably an art.

On the second repertory grid, when mathematics is replaced by three of its sub-disciplines: fractal geometry, calculus and algebra; the multi-dimensional nature of fractal geometry is further accentuated. Calculus and algebra separate from fractal geometry which falls almost exactly in the centre of the grid. It captures aspects of all the other subjects and is, at the same time, rigid and expressive, subjective and objective, and both a descriptive and problem solving tool. Randy confirms this picture with, "I think it's kind of found, what I'd like to think of as its niche, right there where it is" (RW-INT-D-08b).

Randy's love for fractal geometry is not because he sees it as non-mathematical. In fact,

there's lots and lots of technical mathematics associated with fractal geometry. You can open books full of equations on fractals associated with differential equations, all in symbolic form, difficult to read for the, the non-initiated. It could look as mathematical as anything we normally think of as mathematics. (RW-INT-D-08b)

But, within this technical mathematics there are opportunities for self-expression.

Just because you might have some basic structure in mind doesn't mean that we can't improvise on it and be creative....There is a sense of excitement that just rubs off of it, feeds on itself, and works on both parties, the teacher and the student. (RW-INT-D-07)

Randy wants to introduce his students to this excitement and struggles to find opportunities within his packed curriculum to build toward an introductory understanding of fractals, especially the Mandelbrot Set.

 

Mandelbrot Art

 

Since his first introduction to fractal geometry over ten years ago, Randy has been fascinated by the complex details of the Mandelbrot set. The intricate patterns that make up its boundary continually excite him. From past work he knows that the mathematics involved in this subject is accessible to Grade 12 students. Their course contains the key ideas, composition of functions and complex numbers, but getting to the Mandelbrot set does require some digression from the official curriculum. Randy would like his Grade 12 Enriched class to explore this topic, but he feels pressured to cover the regular course content. Looking back on the project, Randy describes his dilemma and choices.

For weeks since I first decided that I wanted to do it with them I couldn't find the right time. We were into mid-term exams and rushing to complete curriculum and always feeling a little bit behind in the Grade 12 course. So I tried to work it in, in a way I'd never tried before, in little bits over a period of about seven classes. In five of these we actually did something on the Mandelbrot set, and in two we did nothing at all.

On day one, with his lesson on transformations of trigonometric functions completed and 11 minutes remaining in the class period, Mr. Walker switches topics to look at solving quadratic equations. Using the examples, x2-9=0 and x2+9=0, and a highly teacher directed approach, the roots, x = ±3 and x = ±/-9, are identified. Mr. Walker responds to student comments that /-9 is undefined with the suggestion that they can still talk about the quantity, keeping in mind that it is not a real number. The root, /-9, is rewritten as 3/-1 and then 3i, with the introduction of the definition, i2=-1. As the class ends, Mr. Walker assigns for homework the task of calculating values for i1, i2, i3,...i13. (RW-LES-N-06)

Day two brings another rushed mini-lesson on complex numbers. The students provide the solutions for the powers of i calculated for homework and note that these cycle through four values: i,-1,-i, and 1. At the board, Mr. Walker plots these as vectors on a complex plane, and the counter-clockwise rotation around the unit circle is noted (RW-LES-N-09).

Mr. Walker begins his third compressed lesson by asking the class to recall their earlier work with the iteration of f(x)=x2 and pixel rainbows. These ideas will be extended for the case of complex numbers. The class notes that all complex numbers with modulus less than 1 will, when iterated in the function, f(z)=z2, spiral into the origin of the complex plane, while those with modulus greater than 1 shoot off to infinity. Using ideas and vocabulary from their previous pixel rainbow experience, students identify the basin of attraction as a unit disk centred around the origin of the complex plane and the Julia set as the unit circle. As class ends Mr. Walker assigns as homework the task, "trace the path of 0+0i when it is used as the seed value for iteration of the function, f(z)=z2+3+2i".

Although the 10 minute lesson is very rushed, with Mr. Walker essentially lecturing, the students remain attentive. There is a rising sense of anticipation. Mr. Walker is obviously excited by this mathematics and for some of the pupils his enthusiasm is contagious. Stephen, Mr. Walker's strongest pupil, stays after class to talk further, concerning the paths of points within the Julia set (RW-LES-N-12a, RW-LES-N-12b).

When the fourth short lesson begins, Mr. Walker finds that the students are confused and were not able to do the calculations left for homework. Acknowledging the messy nature of the arithmetic, Mr. Walker announces that they will now use a computer program that he has written to perform the calculations and plot the progress of the iterations. For the remaining 5 minutes of class time, functions of the form, f(z)=z2+c, with c taking on a variety of complex values, are input to the computer and the iteration path of 0+0i plotted. Some traces spiral into attractors while others disappear off the screen on their way to infinity. Class ends with the announcement, "Tomorrow we will put all these ideas together and take a look at some beautiful pictures" (RW-DIS-N-16).

Randy is excited as he and I collect the media equipment for the last class in the Mandelbrot set sequence. We are going to watch the video, Nothing But Zooms! (Hubbard, Smith & Staller, 1988) which shows an animated exploration of the details of the Mandelbrot set. Randy finds these images intensely beautiful and tells me, "Every time I watch it I get shivers down my spine" (RW-DOC-N-06). While Randy's pupils do not appear to be quite as passionate about the pictures, they watch the video intently and, when class ends before the program has finished, several students remain behind to catch the final images.

Looking back on the sequence of rushed lesson, Randy is not happy with the results and reports, "I'd say it was entirely unsatisfactory and I was wishing I could do it over again in one or two sessions" (RW-INT-D-06). Despite this negative appraisal Randy has provided an opportunity for his students to experience the art in mathematics.

 

Mathematics as a Creative Activity

 

Randy has "been picking up Scientific American sporadically since high school but hardly ever ran into anything in the Mathematical Recreations that [he] could appreciate". "I couldn't see what mathematics was in that, or how it would be important because I thought you solved problems with algebra. What was all this other stuff about?" (RW-INT-D-11b). All this changed during Randy's year of teacher education when he was "exposed to some neat stuff that really wasn't so terribly difficult" and discovered "that you really could be creative in mathematics" (RW-INT-D-11b).

Today Randy believes that "mathematics can be an enjoyable recreation for anyone interested in puzzles, games, problems or computer investigations....One might call such pursuits mental exercise, which, like physical exercise, has it pleasures and benefits" (RW-DOC-D-07). The pleasure is in the exploration rather then the final product since, "it would seem that the creative element cannot be focussed on practical problems alone" (RW-DOC-D-07).

It is important for students to see the creative side of mathematics for:

It probably seems to everybody in high school, it was my experience, that everything's been done, that mathematics is a finished body of knowledge....And so the students, I think, get the feeling that they just have to catch up. They just have to learn what everybody else has learned and then once they have the knowledge then they're finished. (RW-INT-D-05b)

Randy argues that this image is false. Mathematics is a changing and growing discipline.

The new mathematics, the computer experimentation, and things related to dynamical systems show us a whole different view of mathematics, if not of the world itself, where the questions aren't all answerable. The situation is a lot fuzzier and uncertain and it underlines something that's always been true about math anyway, which is that mathematics is itself dynamic, it is always being discovered, discovered and reinterpreted and investigated. There are lots of unsolved problems and I think it's important to try to give students some of that flavour. Otherwise mathematics is a dead subject and most of them don't see any future in it. (RW-INT-D-05b)

Randy wishes to have students "see there's more to it....You don't just do a page full of problems and you're finished. There are always more questions to ask and more avenues to explore" (RW-INT-D-03). "But it's very difficult, within the constraints of both the system and our own experience, to actually accomplish anything more than just the regular course of study" (RW-INT-D-05b).

 

Supporting Students' Mathematical Explorations

 

Randy's personal experience with creative and entertaining mathematical explorations has led him to provide his pupils with open-ended investigations, but such activity has been generally restricted to the special "Fractal Geometry and Chaos Theory" course that he taught while at Golden District Secondary School.

In my fractals course it's largely that type [open ended investigations] of activity....The course is not nearly as structured because it's not so content oriented. The students enjoy that tremendously, they tell me. In fact, that has to be one of the reasons why I do it this way, not so much that I have an objective that this meets so well as to just have the students enjoy themselves a little bit more. It's a better way to learn mathematics....There's an awful lot of direct involvement, it's not nearly so passive. (RW-INT-D-06)

Since transferring to Northern High School, Randy has been feeling increased pressure to complete the specified curriculum and reports, "I've managed to get myself into a little bit of a time line difficulty this year, so we have to always be forging ahead and there's no time to relax and let things develop at their natural pace" (RW-INT-D-06). During my recent visits to Randy's classes I have not seen his students involved in any major independent mathematical explorations. While there have been group and individual work in class, the tasks have been well defined questions set by the teacher. For the time being, until he feels released from the demands to cover all course content, Randy restricts his promotion of mathematical investigations to the Grade 12 bonus questions and independent extracurricular activities with interested pupils.

After a long drive I arrive in the late afternoon at Northern High School for my sixth visit with Randy, but he is not in the mathematics department office where we had arranged to meet. Looking through the classrooms I find him in the computer lab listening intently to Gary, a Grade 10 student, excitedly explain the workings of a computer program that he has written. Randy apologizes for keeping me waiting, but he just has to see the work that Gary has been doing. I join him for the demonstration.

Although Gary is not in any of Randy's classes I have observed the two of them interact briefly on each of my visits to Northern High School. Randy describes their relationship.

Gary is a Grade 10 student who comes into a French class as I'm leaving my math class in the same room and he is always taking note of either what's on the computer when he comes in, or diagrams or equations that are on the blackboard, and he wants instant lessons on all these things. He finds them fascinating and inevitably the conversation ends with his commenting how cool, or fantastic, or great this stuff is. Words that most of your students don't share with you even if they ever do have those feelings. He's developed an interest in fractals from a few of those things. Asks me endless numbers of questions, but takes some of those ideas and goes a fairly long way with them himself. (RW-INT-D-06)

Over the past weekend Gary has used his home computer to write a program that simulates the Chaos Game and generates a fractal called the Sierpinski Triangle. It is this software that Gary demonstrates for Randy and I. Although Randy has written an equivalent program himself and has used it to perform numerous experiments, he joins into Gary's investigations with the enthusiasm of a beginner. With his past experience he knows the questions to ask to get Gary thinking along new routes. "I wonder what would happen if the jump ratio was something other than one to two?" Gary excitedly replies, "Oh! I can change the program to do that." He immediately makes an adjustment to the code to get a jump ratio of 1:3 and has the computer producing a new plot. The triangle fills in with no apparent pattern emerging. "That's strange!", says Randy as if seeing this result for the first time, "I wonder why that happens?" Gary makes a couple of attempts at an explanation but stops each time when he realizes that he is confused and has made some errors. Randy listens carefully, offering only simple encouragement and coaching, "Yes", "Sounds good to me.", "Why?", and "Are you sure?". When Gary finally comes to a full stop, Randy has a suggestion. "Seems to me it might be a good idea to try out lots of jump ratios and see the types of patterns you get. Then you might be able to figure out why some work better than others." Gary likes this plan and packing his books away announces, "I'll try that tonight and tell you what I get tomorrow" (RW-DIS-N-17).

As Randy and I walk back to the mathematics department office, his words make it clear that Gary is not the only one who has benefitted from the past half-hour.

Gary has been showing an amazing amount of energy and interest in the whole thing, almost overwhelmingly so....It is their enthusiasm that forces me to do things like learn how to program a computer just so I can talk to them....Somebody like that stimulates you. You want to think up good questions for them to work on. (RW-INT-D-06)

 

Case Summary: Subject Conceptions and Teaching Practice

 

For Randy Walker, "in mathematics there's still an essence of the undisputable" (RW-INT-D-05b). This tone of absolutism runs through his conversations about the nature of the discipline and at various times takes on Platonist, formalist, and instrumentalist shapes. Randy presents a Platonist image of an activity that is about finding truth; a body of objective knowledge that we can count on (RW-DOC-D-07, RW-INT-D-03, RW-INT-D-07). Platonism is melded with an instrumentalist view in Randy's understanding that humans have discovered some of the workings of the universe and fashioned this knowledge into a set of rigid rules, tools and techniques (RW-INT-D-05b, RW-INT-D-07, RW-DOC-D-07). Mathematics is a technical language that concisely, through the use of symbolic expressions, deals with the idea of quantity (RW-INT-D-01a, RW-INT-D-07). If we begin with the proper assumptions, Randy has faith in the formalist program to maintain truth and consistency through the careful use of deductive proofs, following the model originally provided in Euclid's geometry (RW-INT-D-01b, RW-INT-D-05b).

The absolutist dimension of Randy's subject conception can be seen in action in a number of the teaching episodes presented in the previous narratives and observed throughout the duration of the study. In the Grade 12 course: observations about trigonometric functions and their graphs were distilled down and precisely recorded as a list of facts (RW-LES-N-01,02,03,05,06, 07), strategies for the solution of trigonometric equations were given as well defined algorithms (RW-LES-N-12,13,20), and properties of the exponential operation were described as the "Laws of Exponents" (RW-LES-N-22). Formal proof methods were demonstrated to and required of students when establishing the validity of trigonometric identities (RW-LES-N-19,21). But, absolutism was not the only theme present in Randy's teaching or conception of mathematics. In both we see the beginnings of a fallibilist view.

Randy is not sure where he stands in the debate between Platonism and constructivism. His recent work and reading in the area of dynamical systems suggests that the world may not run in a mechanical clockwork fashion and that possibly our mathematics does not perfectly reflect truths encoded in the universe (RW-INT-D-05b). The beginnings of a problem solving, social constructivist view are revealed with his comments that mathematics is "very much a human endeavour" (RW-INT-D-05b) which has its origins in efforts to address human needs and problems (RW-INT-D-09c). Reflecting on his own probability experiments, Randy is modifying his formalist position as he expands his definition of mathematical proof and adopts the view that there may be validity in arguments based on the collection of overwhelming evidence (RW-INT-D-05b).

We see these newer, emerging views in action in the teaching episodes related to fractal geometry and probability. Here students conducted mathematical experiments and voiced conjectures based upon the data collected. A more tentative image of mathematical "facts" was presented when pupils were allowed to debate and vote on their choice of answers (RW-LES-N-04,08,10). Mathematics was portrayed as an open, growing discipline with creative and artistic dimensions (RW-LES-N-I, RW-DOC-N-06, RW-DIS-N-17). Randy's problem solving conception of mathematics and more flexible view of proof also made limited appearances in his instruction concerning traditional course content. He expanded the topic of composition of functions to include an exploration of iteration (RW-LES-N-ii). Pupils were encouraged to pursue open-ended investigations in the form of bonus assignments (RW-DOC-D-04, RW-INT-D-03). In several classes the use of small groups promoted student mathematical conversations (RW-LES-N-09,11,12,13,15,16) and intuitive visual proofs were employed in trigonometry (RW-LES-N-11).

While the fallibilist image that Randy is developing with work in fractal geometry is to some extent being carried over into his view of more traditional mathematical topics, the reverse is also occurring; some of his absolutist views are projected onto the newer topics. Randy acknowledges the power of Monte Carlo methods for addressing probability questions, but still holds the view that a problem is not solved until an algebraic proof has been developed (RW-INT-D-05b). When experimenting with the iteration of functions Randy puts an emphasis on terminology as he sees a need for students to develop a precise vocabulary (RW-LES-N-ii, RW-INT-D-03).

In his philosophy of mathematics Randy stands with his feet in both the absolutist and fallibilist camps. Similarly his teaching practice is a mix of traditional direct instruction and features from the mathematics education reform program.

To Table of Contents

To next Chapter/Appendix