Review of Derive 5
David Bowers, Suffolk College, Ipswich IP4 1LT, United Kingdom
david.bowers@suffolk.ac.uk
Description of the software
Derive describes itself as "the mathematical assistant for your PC". As such, it is not a piece of multimedia courseware designed to teach the user mathematics. It is rather a powerful toolkit which allows you to "do" maths at your computer. I shall outline later some implications of this for teaching and learning.
Derive belongs to a group of software programs known as "computer algebra systems". Other well-known examples are Mathematica and Maple. The most powerful feature of computer algebra systems is that they can "do" algebra, calculus and other symbolic (non-numerical) manipulation. They will thus have a similar impact on mathematics as pocket calculators had on arithmetic, allowing factorisation, transposition, differentiation, integration and so on to be carried out at the "touch of a button". For this review, I shall concentrate on the features of Derive which are most relevant to mathematics in the FE and tertiary (16-19) sector.
Version 5 of Derive is designed to run under Microsoft Windows and follows standard Windows conventions. Along the top of the screen are the Menu bar and a Toolbar, each with familiar Windows items (File, Edit, etc) together with specific commands for Derive. The main part of the screen contains the "workspace", which can be toggled between an Algebra Window, a 2-D Plot Window and a 3-D Plot Window. These Windows can also be "tiled" within the workspace area, so that for example graphs can be seen alongside the corresponding equations. Along the bottom of the screen is the Expression Entry Line. Immediately below the Expression Entry Line are two Toolbars for Greek Symbols and Math Symbols, so that special symbols can be included in expressions with a single click. These toolbars can be turned off at any stage to maximise the size of the workspace. (See Fig 1).
The way to use Derive can be summarised as: "First author your expression, then do what you want with it". For example, if you wish to factorise the cubic
x3 - 5x2 + 7x - 3
then you author (ie type) this expression in the Entry Line, press Enter, and it appears on the left side of the Algebra Window. You then select Factor from the Simplify menu, and the result
(x - 3)(x - 1)2
appears on the next line and offset to the right to show that it is machine output.
Similarly, to differentiate
x3.cos2x
author this expression and select Differentiate from the Calculus menu, to obtain
3x2.cos2x - 2x3.sin2x
(See Fig 2)
Actually, things are rather more sophisticated than how I have just described them. Whenever a command such as Factorise, Expand, Differentiate, Integrate and so on are selected, a dialogue box appears with a variety of choices. The default settings usually suffice to produce the expected answer, especially for the level of maths routinely encountered in the FE and tertiary sector, but experimenting with the other options can be quite instructive. For example, the Factor command includes, among others, the option to specify the coefficients as complex. (See Fig 3).
The algebra commands available "on a mouse click" from the Menu and Toolbar are:
There are many more utility functions which have been built in to Derive to carry out more advanced mathematics. For example, solving differential equations, working with matrices, coordinate geometry, applications of integration, to name but a few. Overall around 500 functions are listed in the Reference Guide, and they range in complexity from
ABS(z)
which returns the modulus of a complex number z, and
PERPENDICULAR(y,x,a)
which returns the equation of the normal to the curve y(x) at the point where x=a, to
FRESNEL_COS_SERIES(z,m)
which generates m+1 terms of the series approximation to the Fresnel cosine integral C(z). (See Fig 4.)
The last example above highlights the fact that Derive can cope with more than enough maths for students in the FE and tertiary sector, and is also a serious contender for undergraduate and postgraduate work.
Further to these algebra features, the second key feature of computer algebra systems such as Derive is carrying out exact arithmetic and high precision arithmetic. Exact arithmetic means that quantities such as surds, logarithms and so on remain as surds and logarithms, unless they are specifically approximated by decimals. Working in exact form is something that takes a lot of getting used to by students at A-level, and this facility can support such work. High precision arithmetic means that the number of digits can be arbitrarily set, subject only to the power of the computer's processor. This can allow meaningful investigations into topics such as factorials or recurring decimals. (See Fig 5.)
The third key feature of computer algebra systems such as Derive is mathematical function plotting. The increased availability and affordability of graphical calculators mean that function plotting is - or at least should be! - second nature to students of mathematics. Derive offers a slick and fully intuitive graphing facility, with all the standard features of zooming, tracing, and so on. The 2-D Plot Window also allows parametric and polar graph plotting. (See Fig 6.) The 3-D Plot Window allows very impressive real-time rotation of the surface (this feature is new to version 5).
The fourth key feature of a computer algebra system is the ability to write your own programs and routines which utilise the inbuilt mathematical "engine". Version 5 of Derive offers a combination of functional and procedural programming styles. However, probably only the more experienced user would wish to dabble in this.
For those already familiar with the earlier Windows version of Derive (DfW4), these are some of the important new features of Derive 5:
How this resource can be used
There is a growing body of literature (see the References below) which seek to dispel the view that using a computer algebra system in the classroom is somehow encouraging students to "cheat", and discuss how best to deploy computer algebra systems such as Derive to enhance more traditional teaching and learning. The following five examples seem to typify accepted good and innovative practice.
It is my experience that students taking the big step to A-level or equivalent post-16 mathematics can easily be overwhelmed by the plethora of new results and techniques in algebra and calculus which they are expected to learn. They often carry out the methods by rote, with little real understanding or feeling for what is going on. A carefully structured set of Derive exercises prior to a formal presentation of the topic can give students a greater insight and perhaps even enable them to "discover" some results for themselves.
For example, ask the students to use Derive to simplify
log(5) + log(3)
The Derive output (working in exact arithmetic) will be
log(15)
Set a variety of similar, well structured questions. The student should soon be in a position to hypothesise the rules of logarithms. Then a formal treatment (and proof) of these in class should be much more meaningful, since the underlying result has already been "internalised" by the student during the Derive investigation.
The advantage of a computer algebra system for this approach is threefold. Firstly, the exact arithmetic makes the results clear (I have seen students working on a similar investigative approach using only pocket calculators, and being expected to identify that
0.698970004... + 0.477121254... = 1.176091259...)
Secondly, Derive is a tireless provider of such examples, so that the student can generate as many instances as necessary to identify and verify a rule. Thirdly (and this can be a disadvantage rather than an advantage if not handled properly by the teacher), Derive will occasionally throw up "food for thought", for example by correctly but unexpectedly simplifying
log(2) + log(8) as 2.log(4)
When setting work which relies on the features of a computer algebra system, the teacher should always be aware that the software can sometimes be "too clever", and structure the activities to preclude this.
Another example of an investigational activity which can encourage greater "ownership" of the mathematics is to ask students who only know the basic results of differentiation to use Derive to differentiate a variety of products of the form
xn . sinx
for different values of n. I have found that students very quickly "discover" the Product Rule for themselves, and feel much more at ease when subsequently using it.
The creators of Derive claim that it "eliminates the drudgery of performing long and tedious computations". There is no longer a need to set assignments to students which have artificially simple parameter values, just so the student can have a chance of solving them by hand. More relevant real-world problems can be set, and the emphasis placed on modelling the problem and interpreting the results, with the actual computations in between (such as differentiate, integrate or solve) delegated to Derive.
As an example of this, there is a well known problem of finding the maximum length of a pole which can fit round a corner (the corner is where two corridors of possibly different widths meet at right angles, and the pole is constrained to remain horizontal). This problem culminates, after much simplification, in a quartic equation to solve. This would probably defeat most FE students, even if the widths of the corridors were artificially set to "easy" integer values, let alone more realistic ones. But with a computer algebra system to perform the "donkey work", a result can be achieved.
I do not consider this to be "cheating". The student still has to model the original problem, decide a strategy for solution, and verify the answers. Arguably these are higher-order mathematical skills than routinely differentiating a complicated function or solving a messy equation. The intelligent human remains the director of operations, the dumb computer is the slave performing the humdrum algorithms of algebra and calculus.
The "scaffolding" concept was first introduced by Bernhard Kutzler, one of Europe's leading advocates of computer algebra systems in general, and Derive in particular. Mathematics has traditionally been seen as a sequence of hierarchical skills, with the mastery of lower level skills deemed to be essential for the mastery of higher level ones. This can often prove to be a barrier to progression for students, especially in the FE sector where widening participation means students can arrive with non-traditional entry qualifications.
Just as scaffolding is used in construction to allow work to start on higher floors even if the lower floors are not complete, there is a place for scaffolding in education to allow students to make progress with higher level work even if all the lower level skills are not yet fully mastered. In mathematics, a computer algebra system such as Derive can provide this scaffold.
As an example, finding maximum or minimum values of a function analytically relies on solving equations of the form f'(x) = 0. Where the problem concerns volumes, f'(x) will generally be a quadratic, and the technique of solving quadratics is a skill in its own right. So should we forbid an otherwise competent student who cannot prove mastery of quadratic equations from starting a calculus course? Or would it be more sensible to allow the student to get started, rely initially on the scaffold of computer algebra to support him/her over the step of f'(x)=0, and when the time is right come back and make good the lacking foundation skill?
Another example of appropriate use of scaffolding occurred to me recently with a group of part-time engineering students who were using Laplace Transforms to solve problems. The inverse Laplace Transform sometimes requires an algebraic fraction to be decomposed into its partial fractions before the procedure can be followed. The method of partial fractions is traditionally seen as a lower order prerequisite, but these students, who came from industry, had not "done" this topic before. Should I spend valuable lesson time trying to teach this manual skill, for which the students would have no other practical need, or should I allow them to use Derive for this step so that they could swiftly proceed with the problem?
It is seen as a good model of practice, rooted in the Calculus Reform Movement in the USA, to approach mathematics from an algebraic, a graphical and a numerical point of view, and to appreciate the relationship between these approaches. For example, solving a quadratic equation can be done algebraically by factorising or completing the square, graphically by locating the points where the graph cuts the x-axis, and numerically by tabulation, interval bisection or similar techniques. Spreadsheets are deployed by many teachers to illustrate the numerical and graphical approaches, but now computer algebra systems such as Derive can provide the technology to integrate algebra as well.
Notwithstanding the educational benefits that I have outlined above, the main way software such as Derive is used at the outset by teachers and students is to check their work. Have they really integrated that function correctly? Or factorised that cubic properly? Or solved that equation accurately? Even if that were the only use that Derive was put to in a mathematics department, it would be worth having. The novelty to students of having a computer program that will "do" their homework for them soon wears off, in my experience, and students adopt a more mature and responsible attitude. It is certainly beneficial if the students have the means to check their work independently using a computer and correct any errors themselves, rather than waiting until the next lesson for the teacher to tell them what is wrong.
As students' experience of Derive grows, they will realise that there are more sophisticated ways to check their work than simply getting Derive to tell them the final answer. Commands such as Substitute and Declare Variable Domain can be used to see whether intermediate working is on the right track. And furthermore, the classic confusion of students, namely when their "answer" is different from that given in the solutions at the back of the textbook, can often be resolved by Derive. One such way is to author an expression of the form
(student's answer) - (book's answer)
and asking Derive to simplify the expression. If Derive returns 0 (zero), then it is most likely that the student was correct.
Strengths and Weaknesses of Derive 5
Strengths:
Weaknesses:
Overall, the benefits of Derive 5 for mathematics education in my view far outweigh the minor and relatively trivial weaknesses in some aspects of its implementation.
Technical issues
Derive 5 will run on any PC which has the minimal capability of running Windows 95 or higher, or Windows NT. It needs 3 MB free space on the hard drive, and a CD Rom drive for installation. The more powerful the processor and the more memory available, the quicker Derive will crunch through the calculations.
A nice (although rather smug!) feature of Derive is that it reports back in the status bar how long it took to perform each operation. Using my Pentium II 233 MHz PC with 64 Mb Ram (hardly a cutting edge system nowadays!), Derive took 0.024 seconds to factorise the 20 digit repunit 11111111111111111111, and just 2.55 seconds to give the first 50 terms of the Taylor Series expansion of
e-kx.sin(ax)
So it would be fair to say that Derive 5 is essentially instantaneous in its response to anything that a student in the FE and tertiary sector would routinely throw at it.
Derive 5 is provided on one CD Rom, and installs swiftly and easily.
References
Derive 5 is now a product of Texas Instruments. Its "home" on the worldwide web is
www.ti.com/calc/docs/derive5.htmThere are also support pages at
www.derive-europe.comDerive is marketed in the UK by Chartwell-Yorke, who also have a large selection of books on how to use Derive and other computer algebra systems to good effect. Visit
www.chartwellyorke.comThere is a free academic mailing list dedicated to Derive. Browse recent postings and subscribe at
www.jiscmail.ac.uk/lists/derive-news.htmlAnother online discussion group dedicated to Derive can be found at
www.egroups.com/group/edugDetails of the international Derive User Group are available at
www.acdca.ac.at/t3/dergroupAn experimental website with examples of programming for Derive 5 can be found at
http://www.cms.livjm.ac.uk/deriveprogramming/