Some observations regarding the use of graphic
calculators in the USA and Australia
This report is based on some of the issues which surfaced during a trip to the USA in early April, 1998. Travel from Perth to the annual convention of the National Council of Teachers of Mathematics1 (NCTM) in Washington DC was paid for by Hewlett Packard, who had requested that I conduct demonstrations for them in the HP booth in the NCTM Exhibitors’ Hall. Material for this report developed from visits to three schools and also from conversations with the some of the approximately 25 000 teachers who attended NCTM.
Subsequent to the NCTM conference, I visited Gonzaga College High School2 and National Cathedral School3 in Washington DC. During the conference I also spoke to a representative from Bank Street College4 (a New York school) and, while in New York, visited the Nightingale-Bamford College5. I also briefly attended the National Science Teachers Convention in Las Vegas, looking at the use of probes with calculators. Without exception I was treated very kindly by the staff of these schools and I am very grateful to them for their help and their comments.
It should be emphasised that what follows is based on anecdotal evidence and conversations with teachers. It is not supported by any thorough research or formally organised interviews.
Discussion follows under the headings of:
The driving forces…
There was no period in which an attempt was made to write a calculator neutral exam which did not strictly require the use of a calculator, while still allowing students to use graphical calculators.
The AP Statistics exam also requires the use of a calculator (graphic or scientific) as do the AP Physics and AP Chemistry exam. Students must sign a statement if they choose not to take a calculator into the exam and this is not permitted as a reason to challenge a grade.
As in Australia pocket organisers, handheld or laptop computers, electronic pen input or writing pads, devices with QWERTY keyboards (such as the TI-92), calculators with paper tapes, which talk or make unusual noise, or which require an electrical outlet are automatically disallowed.
The AP Calculus exam currently consists of:
Another national system of examinations in the USA is the S.A.T. This examination is undertaken by a far larger population than that of the AP exams. The designers of this set of questions say that there are no ‘graphic calculator active’ questions. This was regarded with some scepticism by some teachers to whom I spoke. The point was made that the power of the calculator makes it very difficult to write any question involving graphing or solution of equations where a graphic calculator does not offer an advantage. The SAT IC and IIC exams are written to require at least a scientific calculator on at least 40% and 60% of questions respectively and graphic calculator are permitted. No other SAT II Subject exam permits calculators.
Overall it seems fair to say that, apart from certain technophile teachers, the acceptance of graphic calculators into the USA educational system has largely been driven, as it has been in Australia, by their increasing requirement in examinations rather than by a feeling that they offer huge educational advantages. This is not to say that the teachers do not feel that they offer educational advantages: it was universally felt by the maths teachers with whom I talked that graphic calculators were ‘the way to go’. Unfortunately, as in Australia, the inertia of the educational system, the conservatism of a significant majority of teachers and (dare I say it) the laziness of some, has meant that many would not have taken up the challenge if not forced to it by the need to prepare students adequately for exams.
One teacher made the point that a part of the reluctance of teachers to fully adopt the use of graphic calculators in the classroom may be partly attributable to their desire to keep their perceived role as the sole distributor of mathematical facts. Some support for this view came inadvertently from another teacher who mentioned that the staff at her school all have a set of twenty or more programs that they keep permanently loaded into their calculators. She didn’t see any reason why the students should not also have access to these programs but this was not standard practice and I received the impression that they were generally used for teacher demonstrations rather than allowing the students to have control of their own learning.
The use of graphic calculators in the USA generally seemed to be phased in at about age 13 to 14 with increasing use over time, increasing strongly at about age 15 to 16. Most teachers choose to introduce functions of the calculator as they become needed pedagogically. It was felt that "maths should drive the machine" rather than the converse. A few teachers mentioned that students in lower years were introduced via less powerful models. The example given was usually of students starting off with TI-81s in 7th or 8th grade and then changing to TI-83s in high school.
Aplets and programs allow the programmer to hide the complexity of the
calculator through a simplified interface. My demonstrations of aplets at
the HP booth provoked a few thoughtful comments from primary school teachers
about "why has no-one written aplets for us?". Nevertheless I feel
that any extension into this area is probably not warranted, since any
applications done in this way would almost certainly be far more effectively
done on a computer offering full multi-media support and the other facilities of
a graphic calculator would be wasted on a primary student.
Training of staff…
Development of expertise can be effectively broken into two aspects. The first level is simply the development of the necessary skills by the teacher to use the calculator effectively as a tool. The second level, for which the first is a pre-requisite, is the effective incorporation of the calculator into lessons as a teaching tool.
The view of all teachers to whom I spoke was that expertise at both levels is developing very slowly in the United States. Opportunities for training were regularly offered but were often not taken up by staff. Most of the major companies such as Texas Instruments, Casio, Sharp and Hewlett Packard offer regular access to teacher training sessions. These sessions often include incentives of free calculators and even prizes of free class sets of calculators. One head of department related how he had been given class sets of HP38Gs on the condition that his staff attend sessions on how to use them and that work was done on adapting the curriculum of the department to use them effectively. This seems an excellent idea and one wonders why this has not happened in Australia. Undoubtedly part of the reason lies with the immensely greater volume of sales inherent in the larger USA population.
One of the teachers who regularly conducted these training sessions for Hewlett Packard remarked that despite these incentives it was often difficult to get teachers to attend, or even to pay attention when they did. He felt that a national exam which required the use of a graphic calculator was still the single most effective issue in promoting growth of teacher skills.
The head of department at one school explained that they have a number of technical assistants, one of whom allocates a quarter of his time just for the math department. In addition to this the department has a one hour meeting each week devoted exclusively to technology - videos, computers & graphic calculators. All department members are also expected to go to at least one conference each year. This relatively wealthy school (with annual fees of approximately Aust.$25000!) offers extensive financial support for staff attending conferences as well as offering staff financial incentives to regularly update their own home computers.
The most commonly used brand in the USA is Texas Instruments (mostly the
TI-83) and so most teachers had expertise in this plus at least one other model.
One commonly mentioned practice was for the staff to specialise in one model and
offer support only for students using that model. Students who chose to
use something else were not penalised in any way but were not given any help if
they encountered problems. One teacher mentioned that her nightmare was of a
student arriving with a calculator with which neither student nor the staff were
familiar. One head of department mentioned that they would be unwilling to
hire any teacher who did not already have expertise in the model used in her
school. This is a point which may well be a problem for future mathematics
The issue of equity is sometimes perceived as being entirely synonymous with either access to equipment or with assessment issues. However, as mentioned earlier it must not be forgotten that access to a teacher who is fully competent may ultimately be a far more important influence on the student’s final results. This point was agreed with by most teachers, very forcefully by some.
Attempts to write questions which are calculator neutral have had varied success both here and in the USA, and the issue is probably better tackled by writing questions in which a calculator can serve as an aid without trivialising the issue. I believe that the best question is one in which a graphic calculator will help the student to experiment but will not give a final solution without a reasonable understanding of the concept.
Aplets and programs…
Most teachers were very happy to see web pages7 appearing offering resources which can be down-loaded. One pointed out pessimistically that for students to benefit from this they needed not only access to the web but also a teacher who cared enough to find the site and bring it to the attention of their class.
Students are extremely conscious of the disadvantage they would incur if their calculator stopped working in the middle of an exam. The increasing complexity of the operating systems is bringing with it a tendency to occasionally suffer ‘system crashes’ in a similar fashion to a full blown computer. This will only increase with time since it is clear that the distance between calculator and computer is becoming largely a matter of semantics. Although the issue has not, to my knowledge, surfaced in advertising as yet, the stability of the operating system of a model may become a selling point in the future.
A further issue which has already become apparent in testing with graphic calculators is the problem created by students who fail to leave a ‘working trail’ behind them and consequently miss out on potential part marks. This is an issue which can only increase in importance with the increasing power of the models. My solution so far has been to ask the students to record as their working trail the equations they entered and the key commands that they used "as they typed them in". This does work quite well since it allows me to spot errors such as those caused by a bracket left out but I am very concerned at the amount of working time which this method steals. The problem is also exacerbated by the fact that the weaker student who is most likely to need these part marks is the one who also most needs the time.
Many seemed to have given it little thought and were surprised that we saw it
as so important in Australia. Some bluntly declared that their students
were not doing it but I suspect that they may simply be unaware of it. One
teacher stated that their school had an honour code of ‘no notes in tests’
that the girls were expected to obey. Unfortunately I suspect that this
would not be a viable solution in the long term due to the dangers of a student
challenging grades on the basis of an unfair advantage to others in her group
who had not obeyed this honour code.
Overall it seems that the issues confronting the USA educational system differ in few important details from those of many states in Australia. More particularly, in those parts of Australia which have adopted graphic calculators, we seem to be at least as advanced in most areas as in the United States.
The far greater population of the USA means that the calculator companies are prepared to invest greater resources and this larger population also means that the USA has the additional advantage of a greater number of innovative teachers tackling the issues. The publication of resources and papers on the web has meant that teachers in Australia have also benefited from the expertise of many of these innovators.
Conversely my experience of the introduction of the graphical calculator into the educational system in Western Australia is that we seem to have proceeded at a slightly more forceful pace. The almost immediate adoption of graphic calculators into the examination system in Western Australia has forced teachers in general to acquire a degree of skill which seems to have been a little slower to develop in the USA.
Of all the issues, the development of teacher expertise is the one which is probably most urgent. Anecdotal evidence, both here and in the USA, suggests that some students are currently being seriously disadvantaged through being taught by teachers who do not know how to use the calculator effectively as a teaching tool or, far worse, do not know the capabilities of the calculator well enough to advise their students on how to use it effectively in examinations.
This issue of teacher training is one which will probably diminish only marginally with time. Training at the student teacher level by universities will help, but the continual release of new models will require some degree of constant updating of skills by teachers in the same manner as that which is expected, for example, in the medical professions. If schools begin to make knowledge of particular models one of their employment selection criteria then the universities will need to ensure adequate preparation of student teachers.
Another consequence of this may well be that market shares by calculator companies will vary with time more than might otherwise be expected, since the release of each new model will offer renewed opportunities to capture the enthusiasm of teachers. The largest market share may go to those companies which are prepared to consistently offer a generous degree of teacher training, since teachers are most likely to recommend to students models with which they are most familiar.
This issue of training can be tackled in a number of different ways:
Isolated examples of collections of calculator resources have begun to appear
on the web, such that of the A.A.M.T. site9, but these collections are not as yet
extensive nor do they offer complete coverage of all models. Pressure upon
such governmental and professional bodies to recognise this responsibility and
to supply the funding and expertise to establish such collections needs to be
exerted by teachers.
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