A number of recent and older papers (Baxter, 1991, Brooks, 1987, Driver & Oldham, 1986, Hand & Treagust, 1991, London, 1990, Stanbridge, 1990, Steffe, undated) describe implications of a personal constructivist theory of knowledge for curriculum development. The approaches described have in common the idea that, since knowledge is individually constructed by learners on the foundation of their prior knowledge, it is impossible to specify uniform learning outcomes for all students. The theoretical basis of this approach to curriculum design is elegantly described by Driver and Oldham (1986). Instead, constructivist curriculum developers flexibly specify (a) the learning experiences in which learners will be engaged, and/or (b) the questions for which they will collaboratively attempt to construct creative solutions. I believe that matrix technique offers a means of tying together these two approaches: to clearly describe problems which the students and teacher will collaboratively attempt to solve, and to suggest possible learning experiences leading toward a solution or solutions.
Matrix technique is also intended to facilitate student synthesis of social and scientific issues, and high level critical engagement with texts. These will not usually be textbooks, but may include written texts such as that in the example given here, videos, newspaper articles, television programs, student written reflections, or any other stimulus material suggested by the teacher or students.
The matrix approach was developed from a constructivist perspective with an emphasis on the social, as well as individual, construction of knowledge; related perspectives include those of Solomon, (1987), Goodson, (1990), Cobern, (1993) and Taylor, (1994). Critical discussion, both in small groups and as a whole class, was identified as a key mechanism for students' construction of new knowledge.
Matrix technique may be defined as the flexible, artistic and collaborative use of the combined problems and matrix, by the students and teacher, for the social construction of learning experiences and new knowledge. The problems may arise from school syllabi, student questions or other sources, or may be suggested by interesting texts. They will generally be in the form of high level questions, propositions for debate, or case studies with related questions. The teacher then develops a matrix; a grid of 16 or 20 words or phrases which suggest possible learning experiences relevant to constructing solutions for the problem. The power of the technique lies in complex and resonant interactions among students, between students and teachers, between the matrix items and problems, and between matrix items. The students and teacher construct the learning experiences together. Jardine and Clandinin (1987, p. 477) describe their model of teaching as story telling: "the meaning, sense and significance of features of the child's curriculum emerge out of the child's sense of the ongoing narrative of the classroom, a shared narrative between teacher and child... The meaning emerges out of the living context of the class, the living history of the class." Within a matrix technique curriculum approach, meaning is not imposed, defined and implemented by the teacher and absorbed by students, but emerges from the collaborative processes of directed inquiry: the 'story' of the classroom.
The matrix items are initially suggested by the teacher on the basis of his or her greater experience of the world, but are open to negotiation and change. The class constructs new meanings and knowledge through its social processes, rather than absorbing old knowledge and meanings from textbooks and authorities. An example of a matrix technique discussion unit on random breath testing, based on a text by Ian Rae (1984) is shown in the appendix. This unit demonstrates the way in which matrix technique takes a written text and uses it as the focus of learning experiences, yet the text is not what is learnt. Indeed, what is learnt cannot be readily specified in advance, since it will be constructed by the students and teacher in the course of the lesson. The learning will depend strongly on the individual context, and will differ for each class. Do some students have experience of driving while intoxicated? Have any lost family members to such drivers? Will the issues of drugs other than alcohol come up? Will the students accept the opinions of the author, or raise questions? Will the legal, ethical or scientific issues catch the interest of the class? Some teachers may find such an open ended approach threatening: others, I believe, will be excited by its potential for organising the social construction of learning in the classroom.
Teachers working from a constructivist perspective will readily recognise ways in which such a unit can be used to allow learners to meaningfully engage some quite complex and difficult moral, ethical, medical, chemical and legal issues, and to synthesise ideas from each of these domains to understand 'real world' problems. In my own teaching, I have felt that such issues were often 'tacked on' to an essentially discipline based study, rather than genuinely integrated. Matrix technique allows teachers to design units which more realistically reflect the complexity of life, through introducing a variety of issues as matrix items. Matrix items do not necessarily reflect the teacher's own ideas of truth; negatively phrased items can serve as valuable catalysts for discussion. The items are also not a closed set: students are welcome to add new items on a consensus basis, as needed to extend the discussion into their own areas of interest.
The use of the unit is intended to go well beyond teacher directed, whole class, verbal discussion, and to organise for the students activities including practical investigations, guest speakers, excursions, small group discussions, and consideration of statistics, newspaper stories and other texts. I include all of these activities in the term 'discussion'. These are only a few examples: I'm sure you can think of more! The organisation provided by matrix technique is not organisation of students' activities and behaviour, but of their ideas and knowledge.
"Constructivist approaches to curriculum development and delivery value and encourage student and teacher thinking, the very sort of thinking we claim to want of our students but stifle with the managerial busy work of competency based curriculum models." (Brooks, 1987, p 67)The combination of the problem with the matrix items suggests, not a linear path to the 'correct' answer, but a wide-ranging yet focused discussion of complex issues. Matrix technique is also intended to facilitate class discussion of what constitutes adequate and relevant evidence for an assertion. As claims and counterclaims are made and evidence is adduced, one role of the teacher is to draw attention to the different types of evidence (moral/ethical, statistical, chemical, opinion) which are raised, discuss their logical compatibility, and say How do you know? On what do you base this idea? In my experience, students will often do this too: when engaged in discussion, they have a very keen eye for the inconsistencies and illogicalities of others' responses! It is important, too, that the teacher accept a role as 'referee' to ensure fair play during discussions: all students must feel safe and secure enough to be able to contribute to, and enjoy collective ownership of, the collaborative construction process. In addition to the unit shown in this article, I developed (Geelan, 1994) a Year 11 Physics course in a New South Wales school, based on the preparation of a matrix unit for each of the six content units which make up the core of the NSW Higher School Certificate (HSC) physics syllabus. An example of one such unit is shown below: the matrix and the four or five problems were used as a focus for all of the activities of the unit. These activities were often fairly traditional however, and many lessons, intended as question and answer sessions, became lectures.
In retrospect, one problem with this application of the matrix approach seems to have been that as a relatively inexperienced teacher I was unable to find creative ways to escape the constraints applied by the looming HSC exam. My own newly developed constructivist role expectations for myself conflicted with the students' highly transmissive expectations and views of learning, leading to frustration and dissatisfaction on both sides. As I asked the students to become more active in their learning, they continued to ask me to "just give us the facts for the exam". Roth and Roychoudhury's (1994) study suggests that this is not a unique problem where teachers attempt constructivist reforms.
An early step in any program of introducing constructivist teaching changes, if it is to be successful, must therefore be a concern with the role expectations and views of teaching, learning and knowing held by students. From a critical constructivist perspective (Taylor, 1994), it would have been much more appropriate, and more effective, if I had negotiated the constructivist reforms with the students at the beginning of the course, rather than unilaterally deciding to introduce unfamiliar roles for myself and them, and unfamiliar teaching strategies. My current research, growing out of this experience, is focused on exploring the theories and ideas which students have about knowledge and what it means to know and learn.
A number of different, flexible strategies must be combined if curriculum development is to be truly constructivist in its perspective; there is no single 'right way'. Matrix technique is offered as one possible way of organising the ideas, knowledge, discussion and inquiry of students in science classrooms, with the intention of promoting the individual and social construction of viable knowledge. Its particular contribution is seen to be in the area of facilitating students' critical synthesis of ideas from a variety of disciplines in seeking solutions to relevant, motivating problems from the human world.
The introduction of constructivist reforms should ideally occur through meaningful negotiation of both (a) the learning environment and (b) ideas about knowing and learning with students. Attention should be directed to their concerns about parental and societal expectations, employability and external examinations, rather than to the unilateral 'constructivist conversion' of individual teachers.
2. In describing motion, we use time as one important measure. Choose items from the matrix which you would use to explain the relationship between acceleration, velocity and displacement.
3. Movement in circles and of objects oscillating on springs seem complex, however some concepts of physics make them simple. Choose items from the matrix which help to explain these motions.
4. The description of motion was once (and still is) of military importance in analysing the motion of artillery fire. Discuss the moral responsibilities of scientists for the ways in which their work is used.
5. As mentioned in problem 4, physics can help describe the motion of projectiles. Which matrix items help to explain the motion of a golf ball during a drive?
| 1. Cannon balls strike as fast as they are fired | 2. Time, times and the dividing of time (Book of Daniel) | 3. Scientists are not responsible for how their work is used | 4. Draw a graph, any graph. Find the slope and the area under it |
| 5. Velocity is speed that knows where it's going | 6. God is the Creator of the physical world | 7. The ball rises in a gentle arc and falls in Lendl's court | 8. At what point does an object fly off a piston? |
| 9. Moving in a circle, speed is unchanged but you accelerate | 10. What accelerates most when it's stopped? When is it fastest? | 11. Displacement is distance that knows where it's going | 12. How far will a skate board roll by itself? |
| 13. I need to be in Brisbane by 2 o'clock. Can I do it? Legally? | 14. My odometer is broken - how far have I travelled? | 15. How far is London? Depends | 16. Physics is simply a search for truth |
| 17. If I started braking now, I'd stop in plenty of time | 18. If she's only 2 metres away, why can't I see her? | 19. My motorbike isn't very fast, but it's very quick | 20. Vectors help us to clearly describe motion |
2. What issues of justice are raised by the practice of random breath testing?
| 1. If people can be discouraged from driving after drinking, lives will be saved | 2. The graph describing BAC vs time is not as simple as we had thought | 3. Potassium dichromate breathalyser bags have been replaced | 4. It would be fairer if the law worked in a series of smaller BAC steps |
| 5. The correlation between alcohol in expired air and blood is not perfect | 6. There is currently no known method of quickly reducing BAC | 7. Breath testing is an intolerable infringement of personal liberty | 8. Chemistry consultants who help get drink drivers freed are irresponsible |
| 9. Police officers are not adequately trained to administer a chemical test | 10. Tin oxide is a reliable method of measuring the alcohol content of air | 11. Tests on blood and urine are more reliable than breath tests | 12. A device which tests the driver's breath before allowing the car to start is useful |
| 13. The link between driver BAC and road accidents has not been well established | 14. Chemists should have a role to play in keeping governments honest | 15. Since there is measurable experimental error in breath testing, it is useless | 16. A zero BAC law is unenforceable, since the decay curve is so long |
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Brooks, M. (1987). Curriculum development from a constructivist perspective. Educational Leadership, 44(4) 63-67.
Cobern, W. (1993). Contextual constructivism: The impact of culture on the learning and teaching of science. In K Tobin (Ed.), The practice of constructivism in science and mathematics education. Washington DC: American Association for the Advancement of Science.
Driver, R & Oldham, V. (1986) A constructivist approach to curriculum development in science. Studies in Science Education, 13, 105-122.
Geelan, D. R (1994). Learning to communicate: Perspective transformations of a developing science teacher. Unpublished M Ed thesis, University of Melbourne.
Goodson, I. (1990). Studying curriculum: Towards a social constructionist perspective. Journal of Curriculum Studies, 22(4) 299-312.
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Rae, I. D. (1984). Letter from Monash. Chemistry in Australia 51(9): 228.
Roth, W. & Roychoudhury, A. (1994). Physics students epistemologies and views about knowing and learning. Journal of Research in Science Teaching, 31(l): 5-30
Solomon, J. (1987). Social influences on the construction of pupils' understanding of science. Studies in Science Education, 14, 63-82.
Stanbridge, B. (1990). A constructivist model of learning used in the teaching of junior science. Australian Science Teachers Journal, 36(4), 20-28.
Steffe, L. P. (undated). Principles of mathematics curriculum design. Unpublished document. University of Georgia, Athens, Georgia, USA.
Taylor, P. C. (1994). Mythmaking and mythbreaking in the mathematics classroom. Perth: National Key Centre for Science and Mathematics Education, Curtin University of Technology.
It had to happen, and when it did it couldn't have come at a better time. We had just left the theatre, only driven a few blocks, when a policeman welcomed us to the glare of the arc lamps and a breath of mobile chemistry. I'd had a single glass of Riesling with my dinner, over four hours earlier, and my breathalyser reading was a nice safe 0.01 percent.
The encounter left me with some sharp impressions. First, that potassium dichromate has disappeared and we no longer wait anxiously to see if the orange turns to green. I blew into a machine with a digital read out: I gather that the heart of it is a tin oxide crystal whose conductivity is affected by adsorption to its surface of compounds such as ethanol. The same effect is produced by carbon monoxide and I was warned to expect a reading if I had been smoking (no worries there). Of course, this was just a preliminary. Had my reading been over 0.05 percent, I should have taken a more serious test which would have involved dichromate and a machine known as the Smith and Wesson No. 900. All this I applaud: the evidence that booze contributes significantly to death. injury and property damage in our society is so strong that objections about the infringement of personal liberties are unlikely to prevail.
What does worry me, however, is the way our laws deal with numerical evidence like that we get from a breathalyser. The penalties which apply to convicted drivers rise in steps marked by particular blood alcohol readings. There must be a certain experimental error in these readings my case suggests that it might be +0.01 for the preliminary machine and it could be similar for the Smith and Wesson and so a scale of penalties with more small steps, rather than a few large ones, might give us more justice. Victoria and South Australia have recently introduced a zero limit for P plate drivers. This is a draconian step which is already affecting the target population, drivers in their late teens. Some elect not to drink at all, while others make arrangements to sleep over, so it's not uncommon to see people turning up to a party with a sleeping bag in one hand and a couple of bottles in the other! They'd better be careful when they drive home next morning, however, because their blood might not have returned to zero alcohol content by then. A senior policeman in Melbourne has recently drawn attention to the possibility that a few liqueur chocolates would be enough to put a P plate driver over the (zero) limit. He went on to deplore the role of chemists and lawyers in these arguments shades of Churchill's moaning that war had fallen into the hands of 'chemists in spectacles'.
The growth and decay curves for blood alcohol level have combined with the severe penalty steps to provide some interesting business for chemical consultants. After we stop drinking, blood alcohol level continues to rise as alcohol is absorbed from the gut, and it can be 30 45 minutes before a plateau is reached. During this time, the level is rising and so a driver who meets the breath squad might have an appreciably higher reading in the official test than at the time of apprehension. Our laws limit this delay to two hours but that is more than enough to take a driver past one of the critical numbers which, in Victoria, are 0.05, 0.10 and 0.15 percent. Some drivers have been successful in convincing the courts that the police figure should be scaled down, by use of the driver's personal 'alcohol response' curve, to a time of apprehension level which would bring a lighter penalty. The response curve is established by a chemist using a standard instrument after the subject drinks a standard amount of alcohol (six 7 oz glasses of Victorian beer in one hour) and then pauses for fifteen minutes for the mouth to be free of residual beverage. The resulting alcohol level vs time graph is then presented to the court, presumably accompanied by some evidence that drinking had ceased only shortly before apprehension. Devious, eh? I haven't touched yet on the question of whether blood alcohol content is reliably indicated by the alcohol content of exhaled air. The correlation is known to be good but not perfect another argument for more small steps in the penalty scale. This correlation is often the subject of legal argument, and in England and Wales the police are now going to rely more on blood and urine analysis when bringing evidence to court. I wonder if well go the same way here? We can certainly expect to see the drink driving laws get tougher, and it's interesting to speculate about what might be in store for us. In Sweden, for instance, I understand that a dinner party host is liable for prosecution if one of the guests is picked up on the way home.
The latest idea is to fit a breath analysis device to the cars of serious offenders, so that their cars can be started only after a 'clean' breath sample has passed through it. Presumably the device I saw (more correctly, it 'saw' me!) could be used in this way, but it would be easy to fool it with a pressure pack of fresh air in an unsupervised test. A grant from the NSW, Victorian and Federal Governments has been made to two Sydney researchers Professors Graeme Starmer and Bob Breakspere to develop the gadget. Somehow, I can't see it being used.
There's still a fortune to be made by anyone who finds a way to bring blood alcohol level down again, so that drinkers can drive home after enjoying a communal 'high'. I remember that fructose has been suggested to speed up ethanol metabolism, but I don't think the scientific evidence is very strong on this point. Perhaps a quick shot of liver alcohol dehydrogenase would do the trick How about it, genetic engineers: can it be produced and deployed so that we can make some progress with real social problems?
Kind regards,
Ian Rae
This article by kind permission from Chemistry in Australia Vol 51, No 9. Sept. 1984© 1995. Copyright of this paper is owned by the Australian Science Teachers Association. You may print it out for your private use, but you must request permission for any other uses of the material.