For Jonathan Ode, mathematics is an intellectual activity; one that involves higher order thinking processes rather than the routine application of procedures (JO-DIS-N-31). Doing mathematics means reasoning, communicating and problem solving, and Jonathan's lessons are designed to promote these activities. In all but two (JO-LES-N-18, JO-LES-N-19) of the 20 lessons I observed there was evidence of at least one of the NCTM's first four standards: problem solving, communication, reasoning, and connections. In most classes, students, in pairs or small groups, worked collaboratively on problems or mathematical investigations. While Jonathan played a strong leading role in five teacher centred lessons, he still managed to present reasoning as the core of mathematics. Here in a highly Socratic fashion, concepts were developed through the observation and generalization of patterns.
Graduate work in both mathematics and philosophy has provided Jonathan with opportunities to formally study logic and the foundations of mathematics. This rich academic experience showed through in his ability to articulate his conception of the nature of mathematics. But, Jonathan's mathematical background is not all within the domain of pure mathematics. His more practical problem solving experience in engineering studies and as an electronics technician were also evident in his descriptions of the subject.
During the school semester of this study, Jonathan taught three classes per day: one at Grade 9, the first year of secondary school, and two in the Finite Mathematics, Ontario Academic Course (OAC), for graduating-year students who are intending to pursue university based post-secondary study. The following vignettes of classroom interaction describe segments of lessons from both of Jonathan's courses.
Class participation was observed to be high during the 20 lessons attended. Most pupils appeared to be ready to tackle the mathematically rich problems and projects that Jonathan provided, but there was opposition from some graduating-year students who objected to the intellectual demands made by Jonathan's teaching style. During the course of the semester this dissention resulted in parental complaints and a caution from the school administration. Nevertheless, Jonathan forged on in the face of this opposition and continued to deliver lessons that captured the spirit of the mathematics education reform program.
My first visit to Jonathan's classroom revealed a theme that was to run through the twenty lessons that I observed. One word answers are not acceptable in mathematics. In Jonathan's view a complete answer involves a carefully reasoned explanation and his classroom practices are deliberately planned to encourage such student input.
The Grade 9 students are arriving for mathematics class as Mr. Ode writes a question on the board.
Johnny says, "Integers are easy as long as you remember the rule: two negatives make a positive."
Give an example of what Johnny means and explain why it is true.
Mr. Ode points out the question to each boisterous and noisy group of arrivals and invites their consideration of the problem. Finally all are in their seats and the lesson can begin. Mr. Ode, rather than assuming the usual teacher location at the front of the room, moves to an empty desk at the back of the middle row.
Mr. Ode: "What did we talk about yesterday?"
The class responds collectively with a variety of answers: "Plus and minus, positive and negative numbers, integers, adding and subtracting."
Mr. Ode: "Johnny is suggesting a strategy. What does he mean?"
One pupil provides an answer, using money, bank balances and debts as a model for situations involving the addition and subtraction of integers. This explanation is made orally to the whole class with nothing written on the board. Mr. Ode makes no comment on the answer, just calls for others to expand upon the ideas presented or to ask questions. This process continues until three students have collectively built up a strong explanation and the lack of further suggestions or questions suggests that all are satisfied.
Now it is on to the usual first task of traditional mathematics lessons, taking up the homework, but here again the style departs from the norm. Mr. Ode, still seated at the back of the room, calls for volunteers to put solutions for six homework questions (JO-LES-D-01, Bober et al., 1988, p. 44) on the board. Students are eager to contribute. Solutions are quickly written on the board and the writers return to their seats. Now Mr. Ode calls for a second wave of volunteers to check the answer and explain how a problem was solved. As the solution to each question is explained, those at their seats ask questions or offer help whenever a presenter is stuck. Through all this, Mr. Ode's only words are repeated reminders, "Be polite", "Help out". With much cooperation and mutual support all six questions are completed and the class breaks into spontaneous applause for the presenters. (JO-LES-N-01)
Students can contribute to the collective construction of mathematical meaning only if teachers provide spaces for their input. When an instructor stands at the front of a class, delivering examples and explanations, there is little encouragement of student mathematical conversation. As in the lesson just described, Jonathan regularly steps out of the spotlight and creates opportunities for the pupils' suggestions.
Jonathan does not abandon his authority as a mathematician and teacher. He has definite plans for his lessons and knows the mathematics that he wishes to develop. His choices of instructional approach are based upon an image of his subject, one aspect of which is that mathematics involves reasoned communication.
Reflecting back on the above lesson, Jonathan provides the thinking behind his approach to the perennial problem of correcting the homework.
Taking up homework is - can be one of the most useless things we do. I think in most cases it becomes a spectator sport, you know, students sit and they watch and at best they hope that they never get called on. So to make homework, putting homework on the board, of any value, what I try to do is to get at different aspects of learning. And of course one of them is communication. Putting the homework on the board itself is communication, written communication, and a lot of what I talk about when the homework goes out, is the form of that communication so that it can be more effective. But again just putting it on the board is only one dimensional because all we're really doing is looking at it. We're not peering [vocal emphasis] at it. By calling on another student, not the student that put it up on the board, to go up to the board and explain the other student's problem, we're creating a whole new environment. First of all, a student has to read another student's work and interpret that, and then describe that interpretation so that other people can understand what that student thought the first person meant. So we're actually on a couple of different dimensions at one time when we do that. (JO-INT-D-01)
Jonathan's writing about the nature of the discipline makes it clear that mathematics is communication; communication between individuals, communication of reasons along with answers. "In other words, a necessary condition for an activity being called mathematics is the ability to explain the processes used [bold in original]" (JO-DOC-D-03). Answers alone are not sufficient. "It doesn't take on any significance until they can explain how they got the answer" (JO-DOC-D-03). This theme is further developed in interviews exploring Jonathan's writing. "Mathematics has to be an active process ... Um, doing with understanding. By understanding I mean more than a passive understanding. That is, it has to be an understanding that can turn around and be communicated back" (JO-INT-D-03b).
In Jonathan's conception of the discipline, reasoning and communicating one's thinking are central activities of mathematics. His teaching strategies, as illustrated above and shown in the fifth lesson I observed, described below, are designed to develop students' abilities in these areas.
I and other experienced mathematics teachers have found that senior, university-bound secondary school students, responding to the demands of high grades, often lose their interest in working to understand concepts and demand direct failsafe routes to correct answers. Jonathan's senior classes follow this trend, but he is not about to abandon his goal of developing mathematical reasoning.
Mr. Ode's OAC-Finite Mathematics class has been studying combinatorial analysis and last night had a selection of textbook (JO-LES-D-05, Stewart, Davison, Hamilton, Laxton & Lenz, 1988) word problems for homework. Today the students unanimously report that they could not solve question 9.
In the binary number system which is used in computer operations, there are only two digits allowed: 0 and 1.
(a) How many different binary numbers can be formed using at most four binary digits (for example, 0110)?
(b) If eight binary digits are used (for example, 11001101), how many different binary numbers can be formed?
(c) A binary code is a system of binary numbers with a fixed number of digits that are used to represent letters, numbers, and symbols. To produce enough binary numbers to represent all of the letters of our alphabet (both upper- and lower-case), how many binary digits must be used? (p. 53)
Mr. Ode, after reading the question out loud, exclaims, "Wow! That's a difficult question. I don't know why I assigned that problem. I'm not sure that I can do it!" There is a slight pause for dramatic effect and then Mr. Ode adds, "You all worked on it last night; right? So we should have lots of ideas. We should be able to come up with a solution if we work together." With this introduction Mr. Ode sits at an empty desk, approximately in the middle of the room. "Okay, who will get us started?" When there are no immediate volunteers, Mr Ode adds, "Mary, how about you going up to the board and recording our thoughts?" Mary moves to the front of the room and takes up the chalk. With Mr. Ode serving as chairperson, calling for more suggestions and identifying speakers, a solution is slowly built.
Actually the students do not have major problems with the computations, but they are not getting the answers given in the text. Following some of the text's worked examples, they have been led into misinterpreting the problem and seeing the binary number 0010 as different from 10. Thus for part (a) they have four cases: one, two, three, and four digit numbers, for a total of 30 binary numbers. The students protest that this does not match with the "official" answer of 16.
Mr. Ode is not willing to provide a solution. He thanks Mary for her work as class scribe and, moving to the front of the class to begin a new topic, provides the suggestion, "Remember we are talking about a number system, like our base ten system, only here we have just two digits. Think about how we write our numbers, say three hundred and twenty-five, and you should be able to come up with a different interpretation of the problem." (JO-LES-N-05)
As we can see Jonathan does not give in to student demands for precise algorithms to generate test and examination answers. This is a course in mathematics and thus thinking is required.
Problem solving has a long history in the mathematics curriculum (Stanic & Kilpatrick, 1989) and its importance has recently been re-confirmed in the reform documents from the NCTM (1989) and OAME/OMCA (1993). For Jonathan, problem solving is not just a curricular issue. As he informs us in his words and concept map, the process of solving problems is central to the mathematical discipline.
"In my opinion there are two prominent characteristics [of mathematics]; patterning and problem solving" (JO-DOC-D-03). Multiple connecting lines in Jonathan's concept map (JO-INT-D-12a, see Appendix E) are labelled "problem solving". In discussing his map, Jonathan elaborates upon this emphasis.
I see that [problem solving] as more of what we do in mathematics as opposed to just, you know, going out and using formulas. ... To make that work you have to move into - higher level thinking processes and that's where problem solving comes in because it's not a matter of getting to an answer. It's a matter of, of creating a structure that provides answers and then looking at the reasonableness of those answers and the structure itself to the problem. (JO-INT-D-12a)
Thus mathematical problem solving involves more than generating answers. "Mathematics by its very nature is a higher level thinking process. Mathematics requires some contact with, not just the operations used in getting the desired result, but with the process itself" (JO-DOC-D-03).
With a bit of work, I could teach one of these [Grade 7 or 8] students to perform the power rule as used in Calculus. Utilizing rote learning and a lot of repetitive examples I could train a student to answer the question; "What is the derivative of x cubed?" and that student would supply the correct answer "three x squared" The question is: has any mathematics occurred? I believe the answer is clearly no. The student was simply performing in a preconditioned manner. No mathematics here! (JO-DOC-D-03)
Furthermore, this understanding "must be active, that is, it must be followed by some form of communication" (JO-DOC-D-03).
Jonathan constructs lessons that are specifically designed to address mathematical understanding, communication and "the ability to solve problems",
and by that I mean genuine problem solving when they are given a problem which is sufficiently different from previously learned problems. One that they cannot immediately determine the answer to without first investigating the problem, exploring various alternatives and deciding on the best strategy. (JO-DOC-D-03)
In translating his conception of mathematics as problem solving into teaching practice, Jonathan has, over the years, read the professional literature in search of problem solving activities and ways to provide instruction in the processes involved. The techniques developed by Whimbey and Lochhead (1979), which Jonathan read about almost a decade ago, play an important role in his planning for the problem solving lesson described next.
PS News: A Sharing of Ideas About Problem Solving, distributed by the Department of Chemical Engineering at a local university, is a bi-monthly list of resources and digest of journal articles and books addressing the development of problem solving skills. Jonathan has been receiving this publication for a number of years, filing away those articles with application to his teaching. He shows me Issue 36 (Woods, 1958 Jan.-Feb.), focussing on an article examining the work of Whimbey and Lochhead (1979) (JO-DIS-D-02a). This is the source of the Whimbey-pair method to be used during Jonathan's next OAC-Finite Mathematics class, about to begin in a few minutes.
Class starts with Mr. Ode posing the question, "Why do we go to school?" This initiates a series of Socratic exchanges through which Mr. Ode develops the idea that we need school graduates who can think and solve problems in general, not just repeat a set of fixed facts or procedures. Mr. Ode is obviously committed to this mission, developing the theme at an upbeat pace from his favourite teaching location, the middle of the classroom. The students, while politely participating in the discussion, seem to have less commitment and many appear content to learn the rules and get their credit. A brief story, about Mr. Ode's "friend", a chemical engineer who struggled with design problems until he realized that textbook solutions could not be applied directly to industrial processes, completes the lesson introduction.
Whimbey and Lochhead (1979) identify as a major source of errors, the tendency of pupils to rush through problem solving activities, performing computations with the given data without prior thinking or planning. They call upon problem solvers to vocalize their thinking, to think aloud, and encourage this by having students work in doer-listener pairs. Mr. Ode adopts this strategy to encourage attention to the meta-cognitive level.
The class is divided into pairs and assigned four textbook questions (JO-LES-D-03, Stewart, Davison, Hamilton, Laxton & Lenz, 1988, pp. 52-53). Mr. Ode outlines the approach to be used. One member of the pair, the doer, will solve the problem, telling the listener what is being done and why at each step. The listener will encourage thinking aloud by constantly asking for detailed explanations. After each question is completed the doer-listener roles will switch.
The class eagerly starts into the task. There is considerable interaction within the pairs, but often the listeners take on a coaching role rather than probing for details at the meta-cognitive level. Many still see the task as getting the correct answer and this is brought out in the summary discussion were students agree that they appreciated the help from their partners but objected to how the thinking aloud requirement slowed them down. (JO-LES-N-03)
Although the students' desire for direct procedures to generate answers works against total success for problem solving lessons, Jonathan persists. Problem solving, being a central theme of mathematics, must be addressed. As we will see in subsequent observed lessons, Jonathan provides other structured activities exploring a variety of problem solving strategies.
For Jonathan, the study of patterns is at the root of mathematics. When humans organize their observational data, note patterns and formulate these into a language, mathematics is born.
Mathematics is the outgrowth of a number of human characteristics. We seem to have a need to simplify things of a complex nature, we need to organize things. Another characteristic is the desire to think beyond the immediate empirical data of his perceptual world to attempt to gain an understanding of a more global nature.... It is a mistake to separate mathematics from our perceptual world. There is a rudimentary connection between doing things and thinking about things.(JO-DOC-D-04)
Looking at the history of mathematics, Jonathan sees this human desire to summarize and extend observed patterns as the driving force behind the development of the subject. "I think most people need to have some sort of motivation for doing mathematics, so they probably would be involved in some kind of a process that would be experimental in nature and that would give them incentives to go beyond [the data]" (JO-INT-D-12b). The mathematics that we develop through observation of patterns allows us to move to a more conceptual level and extend our world.
You're taking what you have and you're looking, kind of pushing it together to see what's there, to find the patterns and then the patterns allow you to extend beyond what you have as - as the given data. And, I think that's a lot of what we do is - is we want to say well what if, and that's kind of our - part of our creative drive. We want to extend beyond what we have right in front of us and say well, if this situation varied what would happen. (JO-INT-D-08b)
In Jonathan's vision of mathematics the development of the discipline is collaborative, with the subject falling in the cooperative half of the principal components (PrinCom) display of his school subjects repertory grid (JO-INT-D-04, see Appendix F). As we have seen in the previously described lessons, Jonathan translates his vision of mathematics as an interpersonal activity into whole class discussions and collaborative pair work. This enacting of subject image continues in the next reported classroom experiences where the Grade 9 students work in pairs and groups of four to explore patterns and solve problems.
Jonathan's commitment to an image of mathematics as the study of patterns and problem solving extends beyond his own classroom. He has taken on leadership positions in his school system and in these roles encourages other mathematics teachers to provide students with collaborative investigation experiences.
On my second visit to Western Secondary, Jonathan is running around, gathering materials for his next period Grade 9 class, and performing his Assistant Department Head duties, encouraging his fellow Grade 9 teachers to adopt some non-traditional instructional approaches. He is frustrated. There are so many good ideas and resources in the journals, especially the NCTM publications, but he can not keep up with the reading and his filing system is falling apart (JO-DIS-N-02). Finally Jonathan locates what he is looking for, an article from the Mathematics Teacher (Krulik & Rudnick, 1985) that provides patterning and problem solving activities. We make copies of the student materials for class and copies of the full article to distribute to the other teachers. (JO-DIS-D-02b)
Krulik and Rudnick's activities illustrate how recording data in charts or tables can aid the search for patterns. The authors supply ready made charts for each problem, but this is too much guidance for Mr. Ode's liking. He would like the students to come up with the chart idea, so before the student worksheet pages are distributed the class explores question number one. The problem scenario is acted out. Five game chips, labelled 2, 1, 5, 3, and 6 are put into a brown paper bag, with Mr. Ode showing each in turn to the class. "Now we will draw out three chips without looking." A volunteer draws the chips 1, 5, and 3 and writes these three numbers on the board. "We will add these together and call it the score. What is the score this time?" The student records 9 beside his three numbers. The chips are returned to the bag, another volunteer selects three, and records the numbers and the score on the board near their seat. "How many different scores can we get?" A variety of answers are given. "We have two draws and scores. How might we record our work to help us figure out the possible scores?" A couple more draws and recordings are made before one student suggests that the results all be gathered on one board. Mr. Ode copies the second, third and fourth selection data below the first. "Any other suggestions about how we night keep a record?" A couple more draws are made and recorded below the first four but there is no suggestion to make a chart and look for duplications. Some students are getting restless. Sensing that the whole-class lesson is taking too long, Mr. Ode cuts short the exploration and presents an organizing scheme by adding lines and column labels to produce a chart. Mr. Ode explains how recording information in chart form can make patterns more obvious and possibly help us solve problems. This is the central idea in the problems for today.
Student pairings are determined by drawing cards and matching: 2 of hearts with 2 of diamonds, 5 of spades with 5 of clubs. The worksheet pages are distributed and the pairs get to work. There is strong interaction within the pairs, sharing solution ideas, conjecturing and justifying answers. With ten minutes left in the class, Mr. Ode breaks into the activity and initiates a whole class summary. "What did you do in these problems? How did it help?" Mr. Ode wants to focus on general strategies, but the lively discussion soon turns to a debate over the answer for the first problem on the second sheet.
Helen Chen wants to seed her front lawn. Grass seed can be bought in 3-pound boxes that cost $4.50, or 5-pound boxes that cost $6.58. She needs exactly 17 pounds of seed. How many boxes of each size should she purchase to get the best buy? (Krulik & Rudnick, 1985, p. 696)
A chart, with columns for: numbers and cost of each size of box, total cost and total weight of seed, is put on the board. Five pairs contribute entries to complete the table but there is still debate as to the best buy. Would this mean the lowest price or the least amount of wasted seed? Mr. Ode encourages the debate, asking for reasons for each position, and lets the class end without a definitive answer. (JO-LES-N-04)
During the term, mathematics as the study of patterns and the value of patterning in mathematical problem solving appear as themes for several lessons. Two months after the above lesson the grade 9 class experiences another problem solving activity that re-emphasizes the need for an organized approach and the value of charts and tables. This time Jonathan selects work from the "Number Patterns" section of Get It Together (Erickson, 1989, pp. 132-137), a book of problems designed for groups of four. Here cooperation and sharing are encouraged by providing the essential information on four cards, one for each group member. Each person is responsible for communicating their information to the group and ensuring that any suggested solutions fit their data. The group task in each case (JO-LES-D-17b, see Appendix G) is to develop a process for generating the numbers and to project it forward or back in time. (JO-LES-N-17)
As we can observe, students are expected to be active in Jonathan's classroom. His instructional methods call for pupils to explore, conjecture about, and discuss mathematical patterns. In Jonathan's image of the discipline, mathematics has historically emerged from such human activity and he provides parallel experiences from which his students can develop their mathematical understandings.
While the student activities in Jonathan's classroom: investigations, mathematical reasoning and debate, solving original problems, and collaborative group work, are all found in the recent reform literature (NCTM, 1989; OAME/OMCA, 1993), not all those involved in the educational enterprise support such teaching (Jackson, 1997; Mathematically Correct, 1996). Students, especially those in the senior years, more interested in high grades than mathematical understanding, desire precise instruction and closely prescribed tasks (Colgan & Harrison, 1997). They wish for assurances that studying will generate good test and examination scores. Parents, with high expectations for their children and memories of their own mathematics education, expect instruction in well defined techniques not open-ended mathematical exploration. Past experience in mathematics class has taught many of these students and their parents that mathematics is essentially just a set of rules and thus direct transmissive teaching makes sense. Jonathan, in bringing his different image of mathematics to the classroom, is swimming against a strong current.
When I arrive for my fourth visit at Western Secondary, Jonathan is obviously frustrated with recent events. There have been complaints concerning his teaching. Some of the OAC-Finite Mathematics students, having received rather low marks on a recent test, have been explaining their poor performance to parents. Rationalisations have focussed on the style of Jonathan's lessons. Jonathan has not talked directly with parents, but using information from the school administration, he constructs his version of the student complaints. "He expects us to do all the work and will not tell us exactly how to do the questions." "He does not tell us the correct answers." Citing the previously described (JO-LES-N-05) examination of question 9 from page 53 of the text (Stewart, Davison, Hamilton, Laxton & Lenz, 1988), some students have suggested to parents that "Even Mr. Ode does not know how to do the questions!"
The vice-principal, having received some phone calls, has talked to Jonathan. Yes, he has complete faith in Jonathan's abilities to do the problems. His reputation as a strong mathematician ensures that. And yes, Jonathan is probably using an excellent approach to get students to think and really understand, but please tone it down. Parents, students, and the public expect mathematics to be delivered as a set of rules.
Publicly, Jonathan displays a casual disregard for the complaints and the administration's requests. Such things are to be expected and they don't bother him. After all, with 23 years of service his job is secure and he is not trying to gain points for a climb up the administrative ladder. Still, our talk of teaching and learning that fills the rest of his first period spare are more strained than usual. Jonathan provides defences for his style, citing his work with other risk-taking teachers and contacts beyond the school system with those teaching university mathematics. (JO-DIS-N-03)
The second period OAC-Finite Mathematics class begins with Mr. Ode writing on the board, in very neat printing, a list of objectives for the next unit on binomial expansion; dealing with (a+b)n. The recent complaints are obviously having an effect on planning and teaching style. A particular example is developed through a rather mechanical Socratic lesson. The class is attentive, but participation has narrowed, with most answers coming from one single student. The rule for the r-plus-first term is produced, but still Jonathan strives for understanding. "Don't memorize this. - There is a pattern here. - You know this already and can always figure it out." (JO-LES-N-06)
The criticism of his classroom practice is obviously a setback, but I get the sense that Jonathan will be back to his former teaching style in little time. Mathematical puzzling is too much a part of his own life for him to let it go. As we wait for his next class, the Grade 9's, to arrive, Jonathan shows me an interesting question that he presented to them a few days ago.
Jonathon tells me, "I left the puzzle with them for a few days and then suggested that they look for some patterns and make a prediction. Some of them are still working in it. I want to see what they come up with" (JO-DIS-N-04).
As we have seen, mathematics, according to Jonathan, is constructed by humans as they solve problems and explore patterns, problems and patterns that exist in their experienced world. Thus mathematics is rooted in human experience. From his years in graduate school, where he worked with highly abstract aspects of the discipline, Jonathan appreciates the power of mathematics to formally envision other worlds. Still, as he tells us in the following paragraphs, the starting point for mathematics is personal experimentation.
There are two different sides to math. One is the analytical side and that's where the mathematics is produced out of its own ideas and the other side is the uh, I guess you could call synthetic math, which is the mathematics that's used in the production of certain things or in, in the quantification or prediction of certain kinds of events. (JO-INT-D-08b)
When constructing his school subjects repertory grid Jonathan attached the attributes, conceptual, scientific and uses experiments, to mathematics. The principal components (PrinCom) display of his grid (Appendix F) places mathematics between these constructs and when discussing this display, Jonathan notes a tension in his image of the subject. The discipline is rooted in and built from experience, that is experimental, but it can also be extended through purely mental activity.
Maybe what it's [the PrinCom display] saying though is that there, there are two, two of my images of mathematics, which are - maybe are in fact in conflict with each other. [pointing to the label, 'uses experiments'] More of the, the constructivist approach in terms of concrete. You work from experiments - scientific, and up here [pointing to the label, 'conceptual'] we have the, maybe the more pure mathematics approach. (JO-INT-D-09b)
The conceptual and experimental aspects of mathematics are also highlighted in Jonathan's concept map for the subject (Appendix E). Here, these labels along with "visual" appear in large arrows indicating the routes by which mathematics grows. Jonathan reports that "I tried to make it [the concept map] in terms of what I'm doing as a high school teacher." While the map pictures how an individual's mathematical knowledge grows it also represents the historical development of the discipline. "I think historically it happens in the same way that one may think of the learning development process. You work your way up through the concrete into the conceptual level, so I think that it's natural to do things in a concrete way first" (JO-INT-D-12b).
I have seen lots of teachers who, for instance, will leave out the experimental part. They get up there. they have well-designed, well-planned lessons with well thought out responses in advance and they teach the lesson and pull out the responses that they want to get and then they leave. And they've left out an important part of the process and they lose a lot of students because of that. (JO-INT-D-12b)
In many lessons, including the next to be related, Jonathan follows his image of the development of mathematics and employs an experimental approach, building abstractions from examples.
Jonathan's personal philosophy of mathematics is strong enough to support curriculum planning that runs counter to the sequence suggested by officially approved course textbooks. As we shall observe, he rejects a text's formal abstract approach and, implementing his view that mathematics is constructed inductively, helps his students develop a formula through sequences of concrete examples.
When I arrive for my seventh visit to Western Secondary School, Jonathan is getting ready for his next lesson, OAC-Finite Mathematics. The class has been studying probability and the next topic is hypergeometric distributions. Jonathan shows me the section in the text (JO-DIS-D-11, Stewart, Davison, Hamilton, Laxton & Lenz, 1988, pp. 246-247) where the authors begin the
Suppose we have a outcomes that would be classified as successful outcomes and b outcomes that would be failure outcomes. If a stochastic process of n trials involving sampling without replacement took place, then the probability of x successes in the n trials is given by
The Hypergeometric Distribution
"I could do it by this way, but it does not seem to make much sense. I have never taken a formal course in stats or probability and it's easier for me to understand if the idea is developed from patterns" (JO-DIS-N-11).
As the class begins Mr. Ode explains that they will not be following the textbook exactly for this next section. "We could do it the way the text does, but in the long run I think that you will find it easier to do it this other way. We will come back to the textbook examples later." A month has passed since the parental complaints incident, and Mr. Ode is moving back into his preferred teaching style. Still he sees a need to justify deviating from a "give the rule" approach.
The Socratic lesson that follows is strongly teacher led, but it does demand considerable student attention and thought. Mr. Ode draws a picture on the board; a jar holding five red blocks and three blue. "We are going to draw out blocks, one at a time. We can model this here with this big beaker I borrowed from the science department and these blocks." Mr. Ode holds up a large beaker and adds the eight blocks. "The first thing you do is you reach in and randomly select one of these. How big is your sample space?" The answer "eight" is given. "Let's say we are talking about the probability of selecting a red one." Mr. Ode takes out a red block. "How big is your event space?" "Five", comes back as an answer. On the board, Mr. Ode begins the construction of a probability tree diagram, with two branches labelled R and B. "What is the probability here [pointing to the R]?" Mr. Ode writes 5/8 on the red branch in response to the answer "five out of eight". The red block is returned to the beaker. "Now instead we are going to try to get a blue one. What is the event space, how big?" A student gives the answer "three" and Mr. Ode writes the probability 3/8 on the B branch of the tree. "And there's your first experiment." The blue block is returned to the beaker and Mr. Ode moves around the class with students taking out and retaining blocks. "What happens to the sample space as we perform this experiment?" Some of the students have noted the change from the situations examined during the past few lessons and one responds with, "It decreases because you don't replace what you take out." "Exactly!", replies Mr. Ode.
Mr. Ode leads the extension of the probability tree through three draws, asking for sample and event space sizes and probabilities for each branch. Eventually the board contains the following diagram.
Now the stage is set for a look at a particular case. Mr. Ode continues, "First we need to define a random variable. What do you want red or blue?" The popular answer is red. On the board Mr. Ode writes "X = the number of red blocks drawn in 3 trials". This is the first appearance of a variable. "What is the probability that X is equal to three? What question did I just ask?" A chorus of answers gives, "That there are three red blocks."
Mr. Ode: "Good, we are going to take three red blocks and not replace any as we go along. Which branch are we referring to here [pointing to the tree diagram]?" Again there is a chorus of correct answers; "The top one."
After a brief pause to let everyone catch up with the development so far, Mr. Ode begins a sequence of numerical manipulations. The writing of each line on the board is accompanied by an explanation.
By the end of this sequence the students can see where Mr. Ode is going and help him with the last line. "That's the same as five choose three and eight choose three."
Some students protest that this type of expression will not work in every case and possible problem situations are given; "Not if you have ten trials." "What if we're not looking at all red ones?" This is exactly the lead that Mr. ode is looking for. He takes up the last suggestion. "What if we are looking at two red ones and one blue one in three draws? Let's look at this question next." With the students supplying the details of possible routes through the tree and the probabilities on each branch, Mr. Ode creates a sequence parallel to the development in the first question. This ends in the expression,
Together the class identifies the origin of each number in the expression and Mr. Ode adds the labels: number of reds, number of red chosen, number of blues, number of blues chosen, total number, and total number of trials.
A pattern has been noted here, but does it work in the first case? Mr. Ode raises the question. "What we are trying to do is develop a pattern that would work here [pointing to the work for the second question] and here [pointing to the first question]. But something looks wrong here [again pointing to the expression developed for the initial question]." The numbers in this expression are labelled: total number of blocks, number of trials, five red blocks, number of red blocks selected. "But what happened to the blue blocks?" "We did not choose any" is the response from one student.
Mr. Ode continues, "So in terms of choices that's -." This generates a chorus of, "Three choose zero."
Mr. Ode: "And what is three choose zero?" "One", comes back from the class.
With this information, Mr. Ode makes one final alteration to the expression for the first question, and the probability of three red blocks in three draws becomes;
"Oh!" "Wow!", collectively from the class. Despite the rather fast pace and the complexity of the problems, the students see the pattern. With a short summary discussion they are ready to tackle the questions in the text. (JO-LES-N-12)
With his senior pupils, Jonathan's teaching has shifted slightly as he takes more control of the class and moves away from the use of open-ended investigations, but his lesson still demands much student thought. Rather than receiving the text's magic formula along with an explanation of its use, the class has been involved in the reasoned construction of a generalized procedure.
Since the solution of problems is the driving force behind mathematical development, it is not surprising that the resulting concepts and techniques have wide ranging application. Jonathan acknowledges the utilitarian aspects of his subject, both in its history and instruction. His words, concept map, and repertory grid all give applications an important place, but as he tells us in the following paragraphs, the sequence is important. Mathematics is not just a set of algorithms to be applied in other fields, but is a collection of models and processes that have been developed in the solution of problems. Applications, while important, come after the initial problem solving.
On Jonathan's concept map (Appendix E) all the linking lines terminate in a node labelled "Applications of Mathematics". For Jonathan, mathematics, both as a domain of knowledge and a school subject, can not exist as a purely abstract mental enterprise. It must have some ultimate connection with other human activity. "I think if you took this [the "Applications of Mathematics" node of the concept map] off - if we remove the applications for mathematics you would almost devastate the whole process. So I think we're looking at kind of - a necessary aspect of teaching and learning of mathematics" (JO-INT-D-12b). This link between mathematics and other dimensions of human action and thought is also displayed in Jonathan's school subjects repertory grid (Appendix F). The FOCUS analysis of the grid shows that the subjects most closely allied with mathematics are physics and electronics. In fact, on the PrinCom display, the mathematical end of Jonathan's "mathematical-non-mathematical" construct points directly toward these two subjects. But here "mathematical" is being used in a limited sense. "When I was using the word mathematical in this context I was thinking more in terms of mathematical algorithms rather than mathematics" (JO-INT-D-09b). Jonathan notes that he has produced a second construct, "answer oriented-open ended", almost parallel to "mathematical-non-mathematical". "Maybe those are the words I should have used ... because the algorithmic fits in with the electronics and the scientific and the physics in terms of the mathematics" (JO-INT-D-09b).
And, in my opinion, that's the short-sighted end. It's the focusing in on - on this one aspect, getting an answer. It - it takes away the creative aspect. And so many teachers, most of the teachers that teach mathematics, fit right in there, in my experience, and in many ways they devastate mathematics because the, the kids who - who want to soar can't with these people. (JO-INT-D-09)
Of more interest to Jonathan, is a second meaning for the label "applications", given by "Mathematical Modeling", his sub-title for the "Applications of Mathematics" node of the concept map.
When I talk about applications for mathematics I'm talking about real problems or problems where you can ... create a model ... and then try it and see whether it works or not. If it doesn't work then talk about it. Well how come it didn't work? What can we change to make it better, and so on. That's what I see as the modelling. (JO-INT-D-12b)
This other sense of applications, as stimuli for mathematical development, is important, for "our experiences are the seeds - that's where our mathematics comes from" (JO-INT-D-12b).
Jonathan provides an example of how an application can lead rather than follow mathematics.
If you were teaching probability or statistics, where you get some kids and say okay we're going to go outside and you're going to clock cars and we're going to talk about queuing, but your problem is to design a stop light that is going to make traffic flow better. We're going to go out and make some measurements, and then we're going to come back in and we're going to look at creating a mathematical model that will help us do that problem. And then maybe we'll go out and measure again and see if we were to change this light to work on the basis of your model, what happens to the traffic. (JO-INT-D-12)
Looking for sources of mathematics in problem contexts, means that the boundaries of the subject become ill-defined. A concern for some teachers but not for Jonathan.
I've had that experience in teaching calculus where people question me, "So what are you doing here, teaching physics or teaching math?" And, and that's a mistake to ask that question. The richness of mathematics comes out when you, when you stop worrying about those barriers and you start looking at it as an interdisciplinary thing. (JO-INT-D-08b)
In fact, for Jonathan, good teaching requires that mathematics arise out of experience and problem contexts.
As a teacher I would say it's a mistake to try to teach an abstract concept without having formed a good foundation in experience. Even though kids can learn that way all they do is they learn a bunch of rules and algorithms and they apply them without really understanding what's going on. And it's really not a very meaningful experience for them. And, without that meaningfulness then they'll never really go on and use that in any kind of a meaningful way. So I think in that sense it's very important to ground everything in experience. (JO-INT-D-08b)
Thus, in Jonathan's vision, mathematics does involve concepts and techniques that are useful in areas beyond the subject, but for him this is not surprising given the discipline's origins in the solution of problems. In his teaching, Jonathan provides opportunities for students to construct mathematical models that, while developing new concepts, also link the subject matter to the physical world and to potential applications.
As the school term progresses the Grade 9 work becomes more abstract and by my ninth visit to Jonathan's classroom the students are studying algebra. The passage from arithmetic to algebra can be difficult. What do all those x, y, and n's mean? Jonathan wants to help his students grow into algebra through experiences that give the symbols meaning. The next lesson to be described, shows how he employs manipulative materials, patterning activities, and collaborative work (Bennett, Maire & Nelson, 1988) to help the students build algebraic models for physical objects. Of course, with the blocks, groups, and independent work, there are some risks. Jonathan takes the time to coach his class of eager 14 year-olds, both on the mathematics and appropriate behaviours.
"Today we will be constructing geometric figures from blocks, and then using the construction to generate numbers, and then analysing the numbers to find patterns. Let's look at an example; a nice easy one. I don't want to make it too hard for me!" Mr Ode places an overhead acetate of an example (JO-LES-D-20a) on the projector.
Mr. Ode continues, "A word of caution - I think that it's nice when we can have fun. Just as long as the fun does not get in the way of the learning, I'm all for it. Today we are going to be playing with blocks.
This announcement is greeted with a class chorus of, "Yea!", "Ho Ho", "Yahoo!"
Over the roar Mr. Ode continues his coaching. "We all understand the difference between progression and regression, Right?" Puzzled silence is the class response. "No?" A chorus of "No's" comes back from the students.
Mr. Ode: "Progression - means going forward - regression means going ..." - "backwards". The pupils complete Mr. Ode's statement.
Mr. Ode: "We've all been in kindergarten already - don't want to go back, Okay?"
Mr. Ode returns to the sample problem on the overhead projector. "All of the problems are designed so that you can not answer all of the parts by doing the construction. In other words, you are going to be able to build levels - level one, level two, level three - and at each of those levels you will be able to count numbers. Then it's going to ask you for level ten, and I'm pretty sure that in every case there aren't enough blocks to construct level ten. Let's begin."
With Mr. Ode building the figures and focussing attention, the students provide numbers for the surface area and volume for levels 1 and 2. When multiple answers are given for the level 3 surface area, Mr. Ode switches to more direct questioning. "Let's take a look one at a time. How much on the top?" The answer, "Two", comes back. "And on the bottom?" "Two." "How much here [pointing to one side]?" "Three." "And [pointing to the other side]?" "Three." "All together so far?" "Ten." "Good! Now how much on this face [pointing to the front surface]?" "Six." "And here [pointing to the back]?" "Six." "Good! So the number all together?" "Twenty-two."
With some debate the students supply the answers 28 and 8 for level 4.
Mr. Ode: "What's the surface area at level five?"
Mr. Ode: "Are you sure?"
Mr. Ode: "Why did you say thirty-four?"
Students: "Because you went up by six. There's a pattern."
"Great!" Mr. Ode is obviously happy. The students have noted the pattern in the numbers, but he wants to take this one step further. Holding up the solid model for level 4 and adding a pair of joined blocks he announces, "Notice that you can see it here too. If we add two blocks like this, how many sides do we add? One, two, three ..." The class joins in, "Four, five, six, seven, eight, nine, ten." As some students complete the count others protest. "But you are covering some. You only add six." Mr. Ode likes this line of reasoning and links the solution paths. "So you can look at the patterns geometrically, and you can look at them numerically."
Mr. Ode continues, "Now what is the answer at level ten?"
"Ooh", collectively from the class. A number of answers are suggested but the popular opinion is sixty-four. Discussion leads to the conclusion that 9 sixes must be added to the first level, so the answer must be 9 times 6 plus 10 or 64.
Mr. Ode: "Now - let's go one step further. The next one is interesting, because now instead of saying the tenth level, we are going to say what about the Nth level? That is, for any number N how could I get there?" Several pupils suggest, "Six times N plus ten.", which Mr. Ode records on the board as 6 × N + 10. "Okay, let's check it out and see if it works. If N were one, that would be what level?"
Students: "The first level."
Mr. Ode: "Is the first level sixteen?"
Mr. Ode: "If N is 2, what does this [pointing at the formula] give?"
Mr. Ode: "Is that the second level?"
One student suggests, "We need N minus one.", to which Mr. Ode replies, "N minus one, I'm going to write that in a bracket.", and the expression on the board is changed to 6 × (N - 1) + 10.
Mr. Ode continues, "Now maybe it will work. Let's check the fifth level." The expression now generates the correct answer of thirty-four and all appear to agree that it will work at all levels.
Mr. Ode: "What we have done is develop a formula that allows us to find the area at any level you want to go to. If somebody asks what's the area at the hundredth level, it's easy. How much?"
The steps are quickly repeated with the volume results and then the class divides into groups of four that will work on similar activities to build algebraic expressions for block patterns. (JO-LES-D-20b, See Appendix H) (JO-LES-N-20)
In this lesson the students have followed the concrete to conceptual sequence of Jonathan's vision of mathematical development. Thus algebra does not appear as magic. Both in historical terms and within this lesson, algebra has been constructed for a purpose; the task of providing general models for patterns.
Jonathan rejects the popular image of mathematics as the embodiment of truth. As he tells us in the following paragraphs, mathematics is an evolving discipline with the validity of concepts in flux.
I'm not a Platonist. I don't believe that there's a mathematics that we discover. I believe that we create the mathematics. I mean certainly, for me, if there is a God he's a mathematician, there's no two ways about that. But, I don't believe that it's something that we simply look for and discover.... I think we create it as we go along. (JO-INT-D-08b)
Jonathan rejects an absolutist view of mathematics. On the PrinCom display of his subjects repertory grid (Appendix F) the labels "convergent" and "factual" are found on construct rays oriented 180 away from mathematics. For Jonathan, mathematics is not a fixed body of facts and when describing truth he tells us, "I'm trying to get away from that concept of it [truth] being out there.... It's part of the process that the person doing mathematics is involved in" (JO-INT-D-03b). Jonathan finds that his expanded image of mathematical proof and truth differs from that of many of his fellow teachers.
If you were to ask one of the teachers in my Department, they would probably say that the best place to look for mathematical truth would be Euclidean geometry. After all, Euclidean geometry was all about "truth". The road to mathematical truth was called deductive reasoning. Deductive proof techniques were the only sure-fire ways for determining whether a statement was true or not. The concept of mathematical truth has changed drastically over the past fifty years. The correctness of the answer is now something that must be negotiated between players. (JO-DOC-D-03)
The lack of absolute certainty in mathematics is not a problem for Jonathan. In fact, he finds the "divergent" nature of the discipline, as presented in his subjects repertory grid (Appendix F), exciting. "As soon as you fix the concept of truth you've actually put blinders on because - you've now excluded a whole host of alternatives as being of any interest" (JO-INT-D-03b).
As we see, for Jonathan, the openness of mathematics is a source of excitement. The direction of the subject is not determined by some fixed set of rules. There are choices and interpretations to be made.
Jonathan sees a need to provide students with experiences that present an open view of mathematical truth.
If we restrict the questions to those with specific answers we run the risk of giving the student the impression that there is nothing new or exciting to do in math because everything has already been determined to be true or false.... What we need to do is introduce some experimentation into math, allow the students to play with the unknown and not worry about the correctness of the answer. (JO-DOC-D-03)
In his classes the validity of mathematical statements is determined through reasoned discussion, not through appeals to mathematical authorities. During my fifth visit to Western Secondary School I observed a lesson in which Jonathan encouraged multiple interpretations and a variety of answers.
Mr. Ode's Grade 9 class is reviewing recent work with powers. The questions on the blackboard begin with, "1) Simplify 213×26×2". After writing out three more questions, Mr. Ode returns to the first and casually adds beside it the message, "Hint: the answer is not 220". Now there is some confusion and neighbouring pupils begin sharing their thoughts on the solution to question 1. "Don't the exponents add up to twenty?" "Does he mean to write it in some form other than two to the twenty?" "Are there other ways?" Clusters of students exchange suggestions and soon there are multiple possibilities.
Mr. Ode begins the discussion of question 1 with an admission, "Yes, this does simplify to 220, but are there other ways of saying this?"
"10242", is given as a possible answer.
Mr. Ode: "Good, any other way of writing this?"
Another student suggests, "(210)2." Mr. Ode writes these answers on the board with the word "or" between them.
Mr. Ode continues, "Any others?"
"410" and "(22)10" are suggested and these are added below the first two answers.
The next response of, "85", causes a murmur of concern to ripple through the class.
Mr. Ode: "Oh! Some of you don't agree with this last one. Why?"
One of the objectors presents an argument. "We can write eight as two to the three, and so that answer, eight to the fifth, is only two to the fifteenth."
Mr. Ode writes (23)5 = 215 beside the original 85, and when all seem to agree with the equalities, a line is drawn through this suggested representation for 220.
Mr. Ode: "So if this eight [pointing to the 8 in the line just scratched out] is not correct, what should it be?"
A chorus of "sixteens" comes back from the class.
Mr. ode: "Any other way of writing sixteen to the fifth?"
When "two to the four to the fifth" is given, Mr. Ode writes, "165 or (24)5" below the previous answers. The students are invited to look for patterns in the list and this prompt results in the suggestion, "We can always switch around the order of the exponents and get two to the fifth to the fourth or thirty-two to the fourth." "324 or (25)4", is added to the list.
Mr. Ode pushes the thinking further. "Is that all the possible answers? Are there any others?"
There is a short pause for thinking before a pupil resumes the exchange. "Make more brackets."
Mr. Ode: "How could we do that?"
Mr. Ode invites the student to put his expression on the board and "((22)5)2", is written.
Mr. Ode: "Great! How many variations can we make on this one?"
Two more students come up and write answers of, "((25)2)2" and "((22)2)5", beside the first.
"So we can write two to the twenty in many ways", and with this summary by Mr. Ode the class moves on to look at solutions to the second question (JO-LES-N-10a).
After the lesson, as we walk back to the Mathematics Department office, I comment upon the extensive investigation that came out from question 1. Jonathan responds with, "Yeah, and all that out of my mistake. I meant to give the hint, that the answer is not two to the nineteenth. I expected that some would forget that two is two to the one and would get nineteen as the sum of the exponents, but when I wrote the hint I focussed on the correct answer. I saw the error as soon as I began working with individual students, but since it was sending everyone off to look for alternative answers I decided to take this direction" (JO-LES-N-10b).
Mathematics, for Jonathan, is multi-facetted, and this complexity shows through in his subjects repertory grid and the efforts in its construction. "It was difficult to find things that for me were clear enough that I could say okay everything for this concept fits these subjects and the opposite fits the other ones well" (JO-INT-D-09b). Despite these struggles, the PrinCom display of the grid (see Appendix F) and Jonathan's subsequent analysis reveal a split personality for mathematics.
On the graphical PrinCom display of Jonathan's school subjects repertory grid, mathematics sits midway between visual art, which Jonathan labels as being "creative", and the physics-electronics pair of subjects, where Jonathan sees mathematics as "answer oriented". When mathematics is employed as a tool in other disciplines it has the rather limited purpose of algorithmicly generating solutions to their questions. "The algorithmic fits in with the electronics and the scientific and the physics in terms of the mathematics. You know, you have formulas and you use them" (JO-INT-D-09b). In this sense, "the fact that, 'mathematical', 'uses experiments', and 'scientific' are all clustered, I think for me is a natural. I mean, that certainly brings out one of my attitudes towards mathematics" (JO-INT-D-09b). On the other hand, "mathematics is aligned with 'conceptual' and 'divergent', and 'creative' up here is not too far off. I do believe that there is a connection between all of these elements right here [pointing to mathematics, conceptual, divergent, visual art, and creative] and that is one of the ways I look at mathematics" (JO-INT-D-09b).
I guess what I'm saying is that there are two different pictures, and maybe that's the way it should be because in some situations, certainly, you want to teach it from an experimental hands-on approach and then move from this area [answer oriented] into this area [creative]. Once you get up to here [creative] then you're free. It's not divorced [from the algorithmic] but it's kind of like a quantum jump, and then you jump right back down again. I know that's interesting, very interesting.... I noticed it [the relative position of mathematics] right off the bat and thought that's very interesting and very odd. And I didn't really get a chance to put it together. Now that I look at it, it's not that it's odd, it's correct. (JO-INT-D-09b)
This interplay, between creativity and more restricted algorithmic activity, that Jonathan suggests as a teaching approach, is for him a reflection of the history of mathematics.
In a sense, science drives mathematics but on the other hand, there are those ideas which are extended beyond the experiential, where people are developing patterns themselves. For instance as in group theory new ideas were developed and then they ended up using them in chemistry. (JO-INT-D-08b)
Thus mathematics has two dimensions. On the one side, "it's convergent, when you've got it neatly bundled up, you have a bunch of facts, and it's all there. And the other way it's divergent, it's open-ended, there's no limitations and you're extending constantly" (JO-INT-D-09b).
Jonathan's concept map for mathematics (Appendix E), further amplifies these multiple personalities of the subject. Here mathematical knowledge, both for the individual and collectively for a society, is pictured as progressing via three modes of inquiry, "experimental, conceptual, and visual". The experimental mode, which is linked with "numeracy, where we play with numbers and we try to go from one step to the next" (JO-INT-D-12b), identifies the core relationships and processes of mathematics. The conceptual dimension, through the use of symbols and abstraction, takes us beyond the basic number patterns and facts.
I guess you could define conceptual as being non-factual. Conceptual means that what you're doing is you're either investigating or learning or discussing something about something that has factual content. So you're looking at connections between the facts, and you're looking at overlying patterns, and maybe even a hierarchy of that depending on where you were to take it. (JO-INT-D-09b)
The adjective pairs, "factual-conceptual" and "convergent-divergent" are essentially synonymous, with their construct rays being collinear on the subjects repertory grid. The "divergent" and "conceptual" ends of these constructs pointing directly at mathematics, show how, for Jonathan, an essential component of mathematics, one that separates the discipline from science (physics) and technology (electronics), is its ability to move beyond the experimental data.
Effective mathematical progress integrates all three of Jonathan's inquiry modes.
If you're taking a look at a problem, it would be well worth it to look at it in all three ways: as an experiment, look at a visual counterpart to the problem, and then try to conceptualize this and abstract from it.
Calculus, for instance, employs the visual geometric and moves into the algebraic conceptual area, although a lot of classical calculus tends to be centred or people attempt to centre it entirely in the conceptual realm, more as an algebraic process, and look at the visual part as being secondary. I don't see that. I see those two as, as being equal. (JO-INT-D-12b)
"Often the algebra symbols just get in the way when a picture might show you some symmetry that solves your problem" (JO-DIS-N-06), "but we see all those algebraic rules in the books, without anything else around them, and you don't see any visual support" (JO-INT-D-12b). For Jonathan, pictures are important tools in the development of mathematics, both for his students and for the history of the subject. He laments the de-emphasis of geometry in the school curriculum but, recognizing the cyclic nature of educational trends, hopes for its return. "Some day, some one is going to come up with the revolutionary idea of doing Euclidean geometry in high school" (JO-DIS-N-08).
While Jonathan's conception of mathematics is of a multifaceted discipline, there is no real conflict between the subject's variety of personalities. A concept begins its life tied to and restricted by the nature of its experiential source. Once sufficient abstraction has occurred, human thought and creativity are able to extend a new idea beyond the bounds of its original context. Mathematics becomes a product of the human mind.
Jonathan's teaching provides opportunities for his students to experience creativity within mathematics. "Periodic Pictures" (Nowak, 1987), an extended project with the Grade 9 class, provides an example.
I think the assignment has a number of features that attracts me to it. First of all, it's an extension of mathematics in the area of art, because what they're doing is they're using the math to create a pretty picture or design, and I think that's a nice thing for kids to do. It's something that provides them a lot of pleasure.... what they have to do is to take their number sequence and create a picture, and here's where I encourage as much creativity as possible. (JO-INT-D-02)
The class has been studying rational numbers, in particular how to convert from the fractional form to a decimal expression. Last night's home work, question number 3 of the project, asked for the decimal representation of 1/7, but with a new twist. How could you do this with a calculator that displayed only three digits? Now Mr. Ode wants to use the ideas suggested on the project pages to tackle more complex examples.
"We need to know how to write a fraction as a repeating decimal when the repeating part has a period that is longer than the number of digits in the display of the calculator."
"I'm going to be assigning some neat [vocal emphasis] fractions like one over thirty-one, or one twenty-ninth, or maybe two twenty-thirds."
The enthusiasm and excitement communicated by Mr. Ode's emphasis of the "neat fractions" is contagious and there is a general buzz of anticipation in the class. "Neat!, All right!"
Mr. Ode continues. "You are going to be working out what these repeating decimals are and then you are going to be graphing them!" "You're not going to believe how great these graphs are going to look."
After a pause to let the class settle down and get ready for work Mr. Ode proceeds. "If we took the fraction 1/31, what is the maximum number of digits we could get before it repeats?" This question generates a variety of answers, but with some debate the class agrees that 30 is the maximum number.
Mr. Ode: "Let's check and see exactly what it turns out to be." "We will take 1 and divided it by 31. Right? Now you tell me what I do next."
For the next ten minutes, Mr Ode, using a TI-81 calculator with an overhead projector display, leads a highly active Socratic lesson that develops a recursive technique for generating any number of decimal places. After three iterations the class has a decimal expansion of .03225806451612903225806 and has identified the repeating segment for this periodic decimal.
Now the class sets to work repeating the process for the fractions 1/17, 1/19, 1/21, 1/23, and 1/29. In ad hoc groups of two and three they support each other, providing help, debating next steps, and comparing answers, while Mr. Ode circulates through the class providing assistance to any groups that appear to be lost.
After approximately 10 minutes, Mr. Ode calls the class back together so that they can look at the next steps in the project, how to create pictures of the decimals. Using a rectangular coordinate system drawn on the black board, Mr. Ode expands upon the example and instructions provided in the assignment guide.
"What can you tell me about this, that's interesting about that picture?"
"Mr. Ode's question generates a number of suggestions. "It's the same." "It's like a mirror." "You could flip it."
With a little probing by Mr. Ode these ideas are put together to identify the symmetry in the pattern.
Mr. Ode: "That's what you are going to be doing with different fractions that I'm going to assign to you." "But I want you to do more to show the patterns and the symmetry." "Let me show you some examples."
Mr. Ode puts up some student work from past years. Colour, and patterns have been added to emphasize the symmetry. In some cases the axes have been skewed or bent to introduce new patterns.
A chorus of "Wow! Awesome! Cool! Neat!" indicates the class' approval of the pictures. They are ready to tackle the project themselves.
Mr Ode: "We are not going to be able to do all this in class but I want to make sure you can get started, so today we will work on ordinary graph paper."
Mr. Ode comes around the room and assigns particular fractions to pairs of students and they set to work calculating the decimal equivalent and planning their picture. (JO-LES-N-07)
Later, back in the mathematics department work room, Jonathan reflects upon the assignment.
It has a very direct connection to patterning which I feel very strongly about. That's exactly what you're doing is you're recognizing patterns in the decimal expansion, the sequence of numbers, and you're using those patterns and you're translating that from a numerical pattern into a geometric pattern which means that you're moving from one realm into another one and you're making a connection which is an important thing for these kids to be able to do. Historically I like it because much of what we do here has strong connections to geometry that have been lost over the years, and any opportunity I get to kind of re-establish some of those connections I like to jump at that. (JO-INT-D-02)
Twelve days later when I am again visiting the Grade 9 class, Mr. Ode is collecting the students' period pictures. As each pair hands in their work he has enthusiastic comments and questions. "How did you do it?" "Explain it to me." "Show me the symmetry." Mr. Ode listens carefully to the explanations, always asking for more details. The results are impressive and next day the hall outside the classroom is decorated with colourful intriguing patterns (JO-LES-D-07e,j,m,n, see Appendix I).
Jonathan is pleased with the students' work. He has met his goals, to provide an opportunity for open-ended exploration and creativity, but still keep the independent activity within the course outline (JO-DIS-N-07).
Of course there are restrictions as there are on any creative process. It has to demonstrate the symmetry that is inherent in the fractional picture. And they have to exemplify that in, in the colouring and how they present it. So, they're learning how to, how to create something and then produce it in a form which is presentable, and the presentation of your work, I think, is very important. (JO-INT-D-02)
Ontario mathematics teachers, on provincial surveys (Ontario Ministry of Education, 1991c, 1991d), claim almost unanimously that problem solving is an important component of their subject, but the strength of this belief must be questioned in the face of data from the same studies showing the predominance of teacher-centred instruction. Jonathan Ode is an exception to this pattern. Mathematics as problem solving is a central theme of all his conversations concerning the discipline, and this image is carried into his lessons where pupils are regularly provided with opportunities to experience mathematics as a problem solving enterprise. Of the various images of mathematics identified in the literature, Jonathan's conception of the subject fits most closely the problem-solving or social constructivist position. Jonathan himself applies the "constructivist" label to his vision (JO-INT-D-09b) when analysing his school subjects repertory grid (JO-INT-D-04, Appendix F) and noting his belief that mathematics originates in experiments. This statement is not just simple rhetoric, for repeatedly in his writing, repertory grid, concept map and interviews he reveals a problem-solving, social constructivist stance.
Jonathan states directly that he rejects the Platonist position (JO-INT-D-08b) and, despite locating nodes labelled "Basic Mathematical Facts" and "Algorithmic Skills" at the centre of his concept map (JO-INT-D-12a, Appendix E), he appears to also dismiss the instrumentalist view, writing that "a necessary condition for an activity being called mathematics is the ability to explain the processes used [bold in original]" (JO-DOC-D-03). While acknowledging that mathematics can be algorithmic when associated with electronics and other applications (JO-INT-D-09b), Jonathan argues that a focus on generating answers limits the subject and such a teaching approach destroys school mathematics (JO-INT-D-09). Mathematics is strongly linked to its applications and school programs should develop this connection; not in a instrumentalist manner by teaching procedures and then applying them for practice, but in the reverse direction, starting with an application context and building mathematics from that (JO-INT-D-12b).
Despite his extensive graduate school experience with formal logics, Jonathan appears to also disagree with the formalist view. He acknowledges the ability of mathematics to exist independent of physical referents but still gives the activities of formal abstract mathematics a constructivist tone, stating that "a lot of the work in universities; combinatorics, graph theory and things like that; still take an experimental mode" (JO-INT-D-12b). He sees the ideas of group theory growing in an inductive manner through the observation of patterns within the formal symbols of the subject (JO-INT-D-08b). Both Jonathan's repertory grid (JO-INT-D-04, Appendix F) and concept map (JO-INT-D-12a, Appendix E) present two aspects of mathematics: "conceptual" and "experimental". The conceptual dimension Jonathan links to abstract ideas and symbols, allowing mathematics to be "creative" and "divergent", but he notes that the development of the discipline follows a balanced combination of these two paths (JO-INT-D-09b). The abstract conceptual stage builds on the experimental (JO-INT-D-12b).
Jonathan understands the formal proof processes of pure mathematics, but when looking at mathematics in the larger world, puts himself firmly in the social constructivist camp, asserting that deductive reasoning is not the only route to truth and that experimental investigations can lead to general agreement on the validity of processes and answers (JO-DOC-D-03). Mathematics is collaboratively (JO-INT-D-04) constructed by humans (JO-INT-D-08b), both at the present time and through history. Humans have a need to analyse and attempt to explain the world and out of this desire is born the language of mathematics (JO-DOC-D-04).
As the previous eight narratives of teaching episodes show, Jonathan's social constructivist view of mathematics is carried into the classroom. In each lesson the class structure, tasks, and activities can be seen to reflect at least one aspect of Jonathan's conception of the discipline. Although selected for their illustrative value, these classes were not significantly different from others observed. In only two cases did lessons appear to run counter to Jonathan's subject vision. These, a pair of OAC-Finite Mathematics classes, took place immediately after illness had forced Jonathan to be absent for a day. In an effort to cover two days work and get back on his course schedule, Jonathan delivered full periods of direct instruction (JO-LES-N-18, JO-LES-N-19). Students remained attentive during these lessons but the large number of questions arising in the closing minutes suggested that there was considerable confusion. Jonathan, recognizing the lack of success, ended the afternoon lesson with an apology for the volume of work and delivery style (JO-LES-N-19). In all other lessons the story of mathematics portrayed by Jonathan's instruction was one compatible with his social constructivist philosophy.
Students, either in a whole class format or in smaller groups, collaboratively built solutions to problems. Communication and discussion of ideas presented mathematics as a reasoned activity. Students conducted investigations or experiments, gathered data, noted patterns and through generalizations constructed their mathematics. Jonathan took advantage of opportunities to encourage alternative interpretations of questions and the development of multiple different but correct solutions. Through a link to visual art, mathematics was experienced as an open creative activity. Of the themes identified in Jonathan's subject conception, only that of mathematics as applications failed to find a significant place in the observed teaching.
Jonathan's vision of mathematics originating in application situations did have expression in the Grade 9 activity where students built structures from blocks and developed expressions for volumes and surface areas. Here mathematics was presented as a modelling tool, but the "application" did not match the more realistic situations that he described during interviews (JO-INT-D-12). During OAC-Finite Mathematics classes Jonathan justified the exploration of ill-defined problems and efforts to develop general heuristics, with references to the work of engineers and the design process (JO-LES-N-02, JO-LES-N-03), but problems originating in these settings were not addressed.
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