For many students who have to learn mathematics, the subject is frequently perceived as a matter of blind faith rather than full understanding or sound intuition. Courses and examinations often consist of a set of "standard" questions which are to be answered by applying "standard" methods. As a means of encouraging the identification and classification of problem type, the recall of method and the students’ mechanical accuracy of rote technique, the traditional style of teaching and assessment is probably suitable; as a way of encouraging independence, creativity and conceptual understanding, it is probably not. We no longer need to train a generation of obedient book-keepers.

It is my view that the "method-marks culture" is outdated, and indeed is having a detrimental effect on students’ progress. I am referring to the common practice (in England at least) of giving students partial credit for identifying and attempting to apply an appropriate method of solution, even if the subsequent working is riddled with errors and the final answer hopelessly wrong. Two things disturb me in such a scenario: firstly, the implication that creditworthy mathematics is essentially algorithm-driven; secondly, the tolerance that is frequently shown toward "silly" answers.

Good maths means never handing in a silly answer, to adapt a well-known saying. How can we encourage this? We need to give students the chance to develop an intuitive feeling for their work, which means giving them the opportunity to investigate mathematics in more than one learning environment. We need to encourage a sense of ownership of mathematics on the part of the students, which means allowing them to make discoveries and not present them with everything "on a plate". The DERIVE computer algebra system can facilitate these aims. The examples of investigational activities presented in this paper illustrate one possible approach.

An "investigational activity", as presented here, is more limited and more specifically defined than an open-ended "investigation". It is a structured sequence of tasks which work towards a perceived goal, allowing students to reach that goal independently and "claim" that knowledge or insight for themselves. The intention is that if the investigational activities are completed (either in a timetabled computer lab session or in directed private study) in advance of the formal presentation of that topic by the teacher, then the results and methods will already have been internalised by the students. They should then accept them and understand the motivation behind them more readily than they would if they had been presented as "standard" facts out of context on the blackboard.

The investigational activities all follow a common structure, known as "What was the question?" (WWTQ). Firstly a topic area is briefly introduced or reviewed. Secondly, a set of routine questions is asked, for which DERIVE can confirm the answers. Thirdly (the key point), a set of answers only is given, for which the student is asked to generate appropriate questions. The expected approach is "intelligent trial-and-error", based on any patterns spotted in the earlier questions, with DERIVE being applied to the proposed question to see if it does indeed have the given answer. Finally, the students are asked to explain what they observed, and generalise any results.

The WWTQ activites are constrained to fit on one side of A4 paper, to allow a uniformity of presentation style. Hints and guidance in the use of DERIVE are contained on the reverse of the sheet, to ensure that any software-specific details to not get in the way of the essential mathematics.

Four sample WWTQ activity sheets are presented here. Equation of a Straight Line is motivated by my experience of students becoming so distracted by the standard equations of coordinate geometry that they fail to appreciate and check such fundamental properties such as the gradient (steep or shallow, positive or negative?) or fail to visualise and verify whether a particular point really does lie on the line they have found. By the end of the activity, the significance of the coefficients of y = mx + c should be fully internalised, and the students should have gone some way themselves towards the independent discovery of the formulae m = (y₂ - y₁)/(x₂ - x₁) and y - y₁ = m(x - x₁).

Completing the Square makes use of DERIVE’s algebra capabilities for expanding brackets and collecting like terms. Writing a general quadratic in "completed square" form is someting that (surprisingly?) DERIVE will not do immediately. It is, however, a standard A-level (pre-calculus) technique, and is useful for graph sketching. My reason for preparing this activity was that a variety of algorithms for completing the square exist (the one I prefer differs from that in the set textbook), and students are often confused when told a different one again by a different teacher or library book. DERIVE allows a number of attempts to be tried out quickly and accurately until the relationship between the various coefficients becomes clear, encouraging the students to formalise their own algorithms for later class discussion. As an added bonus, DERIVE’s graph plotting facility allows the students to hypothesise the link between the coefficients of the quadratic and the properties of the parabola.

Differentiating Products applies the WWTQ approach to the well-known calculus rule. The result (u.v)’ = u.v’ + v.u’ is the classic case of an algorithm which is learned and applied by rote, but rarely appreciated. Too often I have been obliged to give a "method mark" to a student attempting to apply this rule but who really seems to have no idea what it is about. (Additional comment: even though DERIVE can differentiate most functions automatically, I perceive there still to be a consensus that students should nevertheless be aware of the product rule in simple cases.) To get the most from this activity, the teacher should either "hide" the Integrate command from the Calculus submenu, or be confident that the students are unaware of the concept of integration at this stage. Asking the students "what was the question" in this situation allows an alternative lead-in to the concept of anti-derivatives.

Finally, Simplifying Logarithms is a short activity in WWTQ format which attempts to convince the students of the three rules of logarithms which they would otherwise probably just learn by rote. I have never really been happy with the ugliness of verifying, say, log 2 + log 3 = log 6 using the 10-digit approximations provided by a pocket calculator. Thank you, Soft Warehouse, for an exact alternative! It is perhaps worth pointing out here that if you try to confirm the general rule log(a) + log(b) = log(ab) , then DERIVE seems rather hesitant. Declare a and b to be positive, and the expected does happen. Scope here for further student investigation?

The philosophy behind the WWTQ activity sheets is to allow students to gain an intuitive feeling for some topics and claim a certain knowledge in advance of their formal presentation by the teacher. This should unltimately be more satisfying than a "this-is-the-method-now-use-it" approach. DERIVE has opened up the possibilities here by allowing the user to experiment and providing feedback in areas such as algebra and calculus which hitherto have often been passively accepted.

If we want the students to carry out the activities independently in computer lab sessions or directed private study time, we must make it totally clear what they are expected to do. For this reason, a common format has been adopted for the activities, both in terms of the content structure and the layout, which the students soon learn to identify. We even photocopy the sheets on blue paper to make them stand out! The consistency of the style and approach of the activities tends to have a settling effect on the weaker students.

The hints on using DERIVE on the reverse of the sheets have been found to be essential. The students will have had some experience of DERIVE already, but they will not yet be totally confident in its use.

The WWTQ style is intended to provide a novel challenge, and to de-mystify the concept of the question in the eyes of the students. Ideally, the students will be motivated by the challenge to follow the activity through to its conclusion. Unfortunately, not all students are so determined or so speedy. However, even if they only manage the first part of the sheet, they will have spent time consolidating the background mathematics by finding answers to the routine introductory questions, and will feel they have achieved something.

My personal satisfaction from working with the WWTQ activities comes when observing the "aha! effect" as a student gradually discovers the rule which links the answer and the question. This is particularly pleasing if it concerns a topic which I previously struggled to convince the students of when standing at the blackboard.

The activities

NOTE: The Activities will appear in a separate browser window, so that they may be printed off for class use.

• Equation of a straight line
• Completing the square
• Differentiating products
• Simplifying logarithms