Background and research
Michael Meagher may well feel that the arguments for use of CAS as an effective tool are well established, but at this point in the book it is useful to reflect on just how well established these arguments are. Given the titles of numbers of papers considering the use of CAS such as "Education or CAStration," "Symbolic Manipulation on a TI-92 - New threats or hidden treasures," and "Indispensable Manual Calculation Skills in a CAS-Environment," it is clear that not everyone in the mathematical world agrees. Indeed an article in the recent issue of Mathematics Teacher stated More than with any other technology, thinking about incorporating computer algebra systems (CAS's) in secondary mathematics raises major concerns (NCTM, 2002).
The research evidence outlined below, considers the use of CAS and is broken down into sections addressing four key issues, enabling the main thrust of available research to be set out, prior to looking at particular teaching sections and consequent assessment.
THE TEACHING OF MATHEMATICS BECOMES MORE INTERESTING WITH CAS
STUDENTS ARE MORE MOTIVATED TO LEARN MATHEMATICS WITH CAS
In 1998 the Advanced Calculators and Mathematics Education (ACME) Review stated that, the Group is firmly of the view that effective use of graphics and CAS calculators can have a number of important benefits. These benefits include improving students' perceptions of mathematics and consequently increasing their enjoyment of the subject and motivation to learn (SCCC, 1998). This was a considered statement made after considerable research from various countries as well as directly by word of mouth from students themselves. After the publication of the ACME review, research carried out in Scotland indicated that A positive impact was noted, acknowledging a motivational effect as well as an enhancement in pencil and paper (PAP) activities and … many students appreciating the use of the calculator in their own time, assisting with homework, familiarising themselves with the interface and in so doing, raising their confidence with mathematics (Macintyre and Forbes, 2002). Students themselves made the following comments: Oh no, better having it all the time. When I used it to check was at home when I didn't have a teacher.
It helped me through the higher; gave me more confidence and did more for me than the '83.
A number of studies support this point of view. One study in the USA remarked, At the conclusion of last year I gave my students a survey regarding the use of CAS in the classroom to see if their perception of doing mathematics had changed. 65% of the students indicated that using CAS gave them more confidence and enhanced their ability to learn math, while no students said it gave them less confidence (Longhart, 2002). In Germany (Thuringia) a recent survey aimed to find out the views the students have on the use of the TI-89 in mathematics courses. The study, conducted over a period of 3 years, sought the opinions of grade 11 and 12 students on such questions as working with the TI-89 is no problem for me, and The TI-89 makes me feel more confident when solving problems. The students in the survey particularly liked being able to work faster and to check results. The students, who indicated a moderate enthusiasm for mathematics generally, showed an overall positive attitude to the CAS. (Schmidt and Moldenhauer, 2002)
Extensive work carried out by the Austrian ACDCA, www.acdca.ac.at, substantiated this change in attitude and motivation. In two very large projects, (70 classes in 44 secondary schools with some 2000+ students), carried out between 1993 and 1998, research was carried out into the teachers' and students' acceptance of the new teaching methods using CAS technology. They focused, among other things, on changes in student's success in learning, on changes in their attitude towards mathematics and science, and on differences in knowledge, comparing traditional teaching methods as opposed to CAS supported methods.
Students commented on a positive approach to mathematics, perhaps more generated by the group approach to problem solving, which teaching with CAS necessitates, rather than the use of CAS itself. Nevertheless students commented very enthusiastically on change in attitudes to maths although I am not a good mathematician, the many various ways to tackle a problem were very interesting, great and motivating for me and Now we are sure that we have a very sound knowledge about that stuff because we have learnt that by ourselves. The global impression of the project was viewed by students as excellent and most wished to carry on with this method in the future.
In conclusion of this section it is notable that we have concentrated on what the students thought of learning with CAS. We know, however, that motivated teachers means motivated students and the change in teaching style that teaching with CAS necessitates. New Work Plans in the Classroom (Kendal, Stacey and Pierce, 2001) provides an interesting insight into what changes are needed and on the move towards more student centred learning required. The international literature suggests that these changes in teaching style and classroom environment are likely to impact favourably on students' achievement and motivation. The leader of the ACDCA project Helmut Heugl summarised by saying:
We did not find such a "Holy Land" when working with CAS, not for the pupils and not for the teachers, but an interesting learning environment for a
Giving the 'last word' in this section to a student: Overall I found it a hard year, but it was very informative. If I had to choose between an 'ordinary' maths teaching and this one, undoubtedly I would take this one (Heugl, 2002).
- more meaningful,
- more interesting,
- and more future oriented mathematics education.
STUDENTS WHO USE CAS ARE AT LEAST AS GOOD IN 'PENCIL AND PAPER' SKILLS AS THEIR TRADITIONAL COUNTERPARTS
There is a widely held perception by many teachers of mathematics that CAS use will hamper development of, or lead to atrophy of, by-hand symbolic manipulation skills (NCTM, 2002). Indeed a highly polarised view in the UK deems narrow pragmatism, and the associated reluctance to analyse and to discuss principles, continues to undermine attempts to understand the lessons of the past and to improve future mathematics teaching in England. In particular, there is a real danger that we may repeat with CAS the errors we made in taking for granted the use of ordinary calculators (Gardiner, 2001).
There are two inherent dangers in ignoring the development of CAS. Firstly, many students will feel that the subject is out of touch with the real world where such tools are used widely. They may thus be put off studying mathematics more widely. Secondly, students who use them are likely to develop an unthinking dependence on CAS to do the simplest algebraic operations. If we accept that there is a case for use of CAS in the learning of mathematics then there is a strong case for taking full advantage of what CAS can offer and accepting the challenge that they present to the school mathematics curriculum and how it is presented. The core of the controversy is less about whether students should use them (CAS calculators), but rather when they should use them… (French, 1998).
Most teachers agree that fluency with symbolic manipulation is vital for progress in mathematics. Indeed it is this very fluency which provides access to general cases and exact solutions. What is less in agreement is the degree of fluency and thus emphasis given to this within the curriculum. A person fluent with a particular algebraic concept or principle has three distinctive and defining characteristics:
We suggest that too much time has been spent traditionally on the second of these and too little time on the first and last (Kissane, Bradley and Kemp, 1997).
- They can decide when it might be useful to use this concept or principle;
- They have the symbolic skills to do it correctly;
- They know what the significance of the end product is.
Learning the lessons of the past, considered classroom use of any calculator involves an emphasis on the development of mental and written methods working alongside critical use of the calculator. It is important to strike the right balance between the development of the necessary computational and algebraic skills, and the use of technology to free students from routine drudgery. The aim should be to let calculators shoulder the burden of subsidiary calculations, hence enabling students to tackle more realistic and significant problems. Teaching should also become more effective because there should be more time to give greater emphasis to the selection of appropriate strategies and interpretation of results (SCCC, 1998). This balance of PAP and use of CAS will be explored in more detail in the main part of the paper.
What is the evidence that use of CAS will lead to the atrophying of basic skills? In short, many years of research have shown this to be untrue. A number of research studies have investigated students' performance on achievement tests that measured computational or procedural skills after mathematics instruction that incorporated CAS. All but one of the fifteen studies that we reviewed found that students whose courses included a CAS did just as well on test items that required computational and procedural skills as those students whose instruction did not include this technology. Of course, we might expect the CAS-using students to do well if they used CAS on the comparison tests. When a CAS was not used on these tests, eight of nine studies reported that although by-hand computations and procedures were not emphasised in the related classroom instruction, students who used CAS's during instruction did as well or better on the procedurally oriented achievement test as those who did not use the technology (Heid, Blume, Hollebrands and Piez, 2002). This is compelling evidence indeed.
Other studies outside the USA have come to similar conclusions.
In 1996/7 in Norway a study was carried out with students using CAS. This was a small study with two groups of students studying mathematics using CAS. For the pupils from the test groups: 41 examination papers - the average mark was 3.90. For the ordinary pupils: 155 examination papers - the average mark was 3.48. These tests included items where only pencil and paper was allowed. The results replicated the findings of the US studies. Even though the number of the participants in this experiment is too low to draw any certain conclusions, does the survey above give us relatively strong signals of what to expect if we start using the symbolic calculator in the mathematics training. There seemed to be documentation for following claims:
(Aarstadt 1997) www.rogaland-f.kommune.no/~strand/symb_calc.htm
- The calculation skill increases.
- Motivation for working with the subject increases.
- The ability of problem solving is stimulated.
- The professional understanding increases, and then it becomes easier to find solutions in professional problems.
In Scotland as a direct result of the (ACME, 1998) recommendations, a pilot study was carried out between 1990 and 2001 to gather 'home grown' information about the use of CAS. Very little if any use of CAS had been made in Scottish schools prior to this study. Three pairs of schools were chosen to have dedicated access to CAS and three schools chosen as 'control' schools, the latter having access to graphing calculators but not CAS. At the beginning of the study all students were given an 'algebraic skills' PAP test. The students were given a similar PAP 'skills' test at the end of the study, with the second test having some common items to the initial test. The students had no access to CAS in the final examination and this had a profound effect on use as the year progressed. 356 students sat the initial skills test and 227 of these could be traced on the follow up test. Having considered the empirical data gathered through the two algebraic skills assessments there is evidence to suggest a difference in performance across the schools. The intervention schools performed significantly better at the end of the study on the matched questions, those repeated from earlier in the year as well as on the new questions that were pitched towards the year 12 topics (Macintyre and Forbes, 2001).
A study similar to the Scottish pilot and conducted in Montana in 2000, concluded that Contrary to popular belief, we found that using CAS to complete basic symbolic manipulations does not erode the retention of basic skills. To the contrary, the Math 3 students that employed CAS retained or improved their symbolic manipulation skills.
On a slightly different note and by comparison, was the study carried out at the US Military Academy (USMA) during the 1999/2000 academic year. This study sought to investigate the impact of the integration of the TI-89 CAS on students' achievement and attitudes. Unlike most other studies which compared users to non-users, this study looked at changes in technology users after integrating a new tool (the TI-89 as opposed to the HP 48G or others without "qwerty") with additional capabilities. Again, unlike other studies, the students in this one at the USMA were permitted to use the TI-89 in the final exam. Once again the results supported the assertion that appropriate use of CAS can enhance the teaching and learning of mathematics and lead to improved procedural, conceptual and application performance (Connors and Snook, 2001).
The fact that using CAS does 'no harm' to students' 'basic symbolic manipulation skills', however, does not in itself provide a complete case for using CAS. Many studies have looked at the effect of using CAS on students' mathematical skills and their understanding of concepts. A range of studies showed that the overall conceptual understandings of the students who used a CAS were the same as or better than the understandings of their counterparts who did not use a CAS. Other studies showed that students who used a CAS better understood particular concepts - variable and function - than those who did not use a CAS. From these studies, we conclude that CAS use enables students to develop deeper conceptual understanding" (Heid, Blume, Hollebrands and Piez, 2002).
Giving the second last word in this section to the ACDCA project, Heugl summarises nicely the advantages of using CAS. Mathematical competence opens doors to productive futures. A lack of mathematical competence closes these doors. But the vision of mathematics teaching and learning is not the reality in the majority of classrooms and schools. In this age of information technology students need to learn a new set of mathematics basics that enables them to solve problems creatively and resourcefully. The National Council of Teachers of Mathematics in USA (NCTM) formulated guidelines for educational decisions laid out in principles and standards. One of these principles is called the "Technology Principle": Calculators and computers are reshaping the mathematical landscape, and school mathematics should reflect these changes. Students can learn more mathematics more deeply with appropriate and responsible use of technology. The investigations of the Austrian CAS projects justify an optimistic point of view. The major results are not especially mathematical contents - it is a more pupil centred, experimental way of learning. In other words, the new tool supports all of the four key qualifications…. CAS-supported mathematical education supports and encourages the 4 key qualifications:
much better than traditional mathematics education (Heugl, 2000).
- Subject competence
- Methodological competence
- Social competence
- Personal competence
The very title of the article "New emphasis of fundamental algebraic competence and its influence in exam situation" opens the discussion about standards and expectations of students and their expected behaviour, particularly in high stakes exam situation. "Indispensable Manual Calculations in a CAS Environment" - what are they? (Herget, et al, 2000).
The last word in this section goes to Australia.
What are the Future Goals for Mathematical Proficiency in the CAS-Age?
If computer algebra systems (CAS) are to be permitted in upper secondary mathematics teaching, careful consideration should be given to the balance between by-hand and by-CAS procedures (Flynn, Berenson and Stacey, 2002).
What should students record when solving problems with CAS?
There are people who believe that the calculator acts like a black box, producing answers without any understanding being required by the students and there are others who believe that a very good understanding of mathematics is required in order to use the calculator (CAS) to solve problems. From our experiences….we would argue that the latter is definitely the case (Ball and Stacey, 2001).
The answer to these two questions is fundamental to the whole issue and leads us to the final section of this research summary.
HIGH STAKES ASSESSMENT AND CAS ARE COMPATIBLE
The answers to these questions may well depend on the country concerned and whether or not the examinations are centralised or not. Whatever the nature of the examination structure as Drijvers says, as soon as assessment gets involved, however, attitudes seem to be less favourable. Teachers are afraid that students will not develop "hand skills" when technology enters the examination room. They may fear that students will pass the examination just by button pressing, and that testing becomes unfair because of different access to new technologies. In many countries a debate is going on about the role of new technologies at examinations. School authorities, examination boards and teachers are struggling with the question of how to deal with these powerful instruments. Different policies are followed and so far there is no convergence (Drijvers, 1998).
The situation has changed quite markedly in some countries with regard to CAS in assessments, and progress has been made in some countries, since Drijvers paper was written. Austria now has a well-developed approach to assessment, albeit different in form from prior to CAS. Countries like Denmark have developed central examinations consisting of two parts, Pencil and Paper (PAP) and an 'anything goes' paper. Slovenia has looked at alternative examinations for the Matura to see if this can be adapted to accommodate the use of CAS by students. (Lokar and Lokar, 2001) In England between 1994 and 1996, working groups were set up to assess the potential impact of CAS on the A-level curriculum and examinations. (Monaghan, 1997) More recently the CAS-CAT project in Melbourne Australia has produced a number of influential papers and solutions to what has otherwise seemed like an impasse.
Yet still countries like Scotland and Germany retain the ban on CAS (at least one state (Thuringia) is piloting the potential of CAS - a start!). The only major study conducted in Scotland found that concerns over the 'Black Box' aspect of technology have been unfounded. Indeed, given appropriate conditions and encouragement, the use of CAS as an investigative tool is something that was well received by the schools in the study. The full potential of such technology however, will not be made apparent until schools are encouraged to utilise the resource through a change in policy on assessment. Regardless of personal visions and expertise in harnessing the power of the technology, staff are going to be discouraged and students themselves will be reluctant to make use of CAS calculators whilst they remain 'banned' in assessments (Macintyre and Forbes, 2002).
The ACME Group (SCCC, 1998) recognised the barrier of assessment during it's deliberations and an early recommendation was to adopt non-calculator papers (implemented in Scotland at ages 16 + from 2000) in order to 'free up' the use of technology. Little progress has been made since then. The Group also recognised that the "assessment tail tends to wag the curriculum dog!" Perhaps it is now time to take a radical look at what we want the students of today to learn in mathematics and THEN find a suitable way of assessing them.
The ACDCA Projects recognised this early on in their deliberations and it is perhaps from them we can learn. Which is the more valid question?
Do the new ways of mathematics learning and teaching influence the exam situation?
Does the exam situation influence new ways of learning and teaching? (Heugl, 2000)
There is little doubt that in the past the exam situation has always had a great influence on the content and the teaching of mathematics education. Previously, the emphasis was on teacher led lessons and process based assessment. Times in mathematics have changed! Assessment must change also!
A fundamental principle formulated by the American NCTM is called the Assessment Principle. Assessment should support the learning of important mathematics and furnish useful information to both teachers and students.
The ACDCA CAS projects recognised that the traditional way of assessment was not suitable to the new ways of learning which they observed in their CAS classes.
- Assessment should be more than merely a test at the end of instruction to gauge learning.
- Teachers should be continually gathering information about their students.
- Assessment should focus on understanding as well as procedural skills.
- Assessment should be done in multiple ways, and teachers should look for a convergence of evidence from different sources.
- Teachers must ensure that all students are given the opportunity to demonstrate their mathematical learning.
Some significant changes of the learning process in our CAS-classes which strengthen the necessity of changes in the exam situation:
(Heugl, 2000) The main section of this paper now looks at these ideas in detail to examine how a number of different countries have developed these ideas in teaching sequences and the consequent changes in assessment which result.
- A more pupil-oriented learning process. More frequently mathematical discussions among the students. The teacher is not the source of knowledge; he supports the independent acquisition of knowledge by the students.
- Experimenting, the trial- and error method: We seldom find the "algorithmic obedience" where the teacher shows one way which all the students then accept and follow
- Working in pairs or groups can be seen much more frequently.
- Beside the teacher there exits now a new, very competent expert - the tool CAS. That means pupils do not always need the teacher for examining the correctness of their ideas and results.
- Phases of "open learning" where the students are individually organising the velocity and the contents of their learning process.
- The CAS is not only a calculation tool, students can also store knowledge by defining modules or using the text editor. Therefore it is senseless to forbid the use of learning media, like books or exercise books during the tests.
- New emphasis on fundamental competence. A shift from calculation skills to other competence and skills.
- A clearer emphasis on problem solving
- A more application oriented mathematics.
- More frequent cross curriculum teaching."
CAS Table of Contents