Practice sessions and tutorial classes in mathematics have seen little change in format this century in many universities. Students usually come to a large classroom, where they have access to a number of tutorial assistants.
A new approach is suggested, which is based partly along the lines of humanities' tutorials, in which the students prepare material before the tutorial takes place. This new approach also has extra features incorporating other ideas. Students learn to self-correct their scripts, to present ideas in front of the class, and to work the practice session as a collaborative group.
Student assignment scripts can now be marked in more detail, yet the time required by the tutor for marking is shorter.
Implications for assessment and computer package features will also be discussed.
``At many big universities, the large thrice weekly lectures in a lower division math course are supplemented by once- or twice-weekly `problem sessions' or `help sessions'. Usually the lectures are delivered by a professor or instructor while the help sessions are staffed by graduate student teaching assistants.'' [Krantz (1993) p.34]
Is the above applicable to your university? Do such ``tutorial'' sessions work? Is there another way?
Such questions had concerned me for a number of years, and forced me into action when I moved to a new teaching environment at Bond University. I had a class numbering just under 100 broken into six tutorial groups. The tutorials were not efficient for every student, and I was sure that the tutorial assistants were having all sorts of trouble.
In fact, I noted how little the tutorial system had changed since I was a student at university some thirty years previously. I remembered that I hadn't found the tutorial system particularly helpful as a student, and had discovered much of my university mathematical knowledge either on my own or in one-to-one conversations with various friends. I hasten to add that the remaining discoveries were made when I had to teach the various sections myself - for that is when you really concentrate on researching and understanding a topic.
This was a workshop or practice-session approach. There were usually a number of tutors in the room, or alternatively students were put into tutorial groups and given a tutor. An assignment or problem sheet was handed out, and students were expected to come to the tutorial and work on a problem. If they had trouble, they had immediate access to a tutor. Not all students were considering the same problem at the same time.
Many students turned up and first began working on a problem in the tutorial session. The session seemed to be a way of saying that this was maths problem time, and tutorial assistants are waiting to explain your problems in this room, instead of you having to track them down in their offices. The tutors essentially did most of the talking.
Then there was a change. Students were asked to contribute something by doing the problem on the board. Some tutorial rooms had boards right around the room, and tutors watched many students up at the boards writing solutions to problems posed. It was like a chess master playing ten lesser players at once, and was a difficult session to handle efficiently. The tutors said less, but what they did say was to only one student effectively at a time.
Three members of my family had just completed Arts degrees before I moved to Bond in 1989. I asked them about the tutorial sessions in their humanities' subjects and was intrigued by their replies. First of all they mentioned ``liquid'' tutorials where the tutors supplied red and white wine to help the students participate verbally. I am not advocating this but the cordial nature of this gesture says a lot for the concern of the tutors in trying to make tutorials work. Secondly, there was the fact that each tutorial was prepared in advance by a different person - always a designated student. The student prepared and delivered a short paper on a topic, and the rest of the tutorial group and the tutor commented. This preparation was worth marks, and the class felt let down if a student failed to prepare adequately. When such cases arose, the tutor had to be ready to take over.
I realise that solving mathematics problems is not quite like discussing a problem posed in a humanities class by adopting and defending a particular point of view, but they are not that different!
So, bearing all this in mind, I devised a different approach to mathematics tutorials.
The course that I concentrated on was a service mathematics course for business students, but the process could be used for many other large freshman or sophomore classes for science or engineering students also.
The students were supposed to attend 3 lectures and 1 tutorial per week for 12 weeks. The thirteenth week is for revision and final exam preparation, while the fourteenth week completes the semester with final exams being held. (The semester at Bond is really a trimester since we have three each year, but the administration doesn't understand the mathematical and literal correctness of using trimester yet!)
I broke up the course into 12 modules (one per week) and prepared an assignment sheet for each module. These sheets were given to the students at the beginning of the course. To illustrate the content of a module, here is a summary of the syllabus.
Week 1. Rationals, irrationals, elementary word problems using algebra. Monomials and polynomials. Linear versus non-linear.
Week 2. Straight line. Shift of axes by translation. Absolute value function and split functions. Week 3. Quadratic expressions and equations. Cubics. Curve sketching using simple analysis and computer software.
Week 4. Inequalities (linear, quadratic and absolute value functions included). Solution using manual and computer sketches. Two-variable inequalities.
Week 5. Linear programming (geometric approach).
Week 6. Arithmetic and geometric progressions. Compound interest and discounting. Exponential and logarithmic functions, including computer sketches.
Week 7. Matrix operations using analysis and EXCEL.
Week 8. Solving systems of linear equations by Gauss-Jordan elimination and matrix inversion. EXCEL applications.
Week 9. Basic differentiation of x^n, e^ x, ln x. Product and quotient rule.
Week 10. Chain rule, local maxima and local minima using analysis and computer sketches.
Week 11. Absolute maxima and minima, indefinite and definite integrals.
Week 12. Partial differentiation. Critical points, local maxima, local minima, saddle points. Lagrange multipliers.
Because my final exam is an open book exam, students need feedback on the tutorial assignments to take with them into the exam room.
Each assignment sheet is broken up into two sections. The first section is labelled Practice, and consists of 5 or 6 easier straightforward questions (P1 to P6) similar to the examples given in the lecture and emphasising each of the main points covered. The second section is labelled Hand-in and consists of 4 or 5 moderate to hard problems (H1 to H5), which are small extensions of the lecture examples. They might combine two ideas, or be a business application involving the mathematical techniques developed. The students are asked to hand in H1 to H5 for feedback, but may hand in P1 to P6 if they like. I use the same assignments each semester.
It is essential from an educational point of view that a tutorial have only one tutor and be restricted to less than 20 students. In this way the tutor can get to know the names of everyone in the group and find out their mathematical background. Then the students become confident about the tutor because they feel that a personal interest is being taken in them. Large practice classes with lots of tutors should be done away with on educational grounds. Reasons involving money, rooms or politics will be given for retaining this old format, but ten semesters of the new approach have convinced me that it is more helpful for the students,and makes them better mathematicians. Here is how it works!
Students bring their Hand-in solutions to the tutorial for the appropriate week. The assignment for the lecture material covered in Week N is due for submission in Week N+1.
At the beginning of the tutorial I hand out one-line answers to the Practice questions, and full model solutions for the Hand-in questions. Students use these to self-correct their assignment, noting where they have made mistakes or produced a different solution. Any questions that cannot be solved are given a big note (asterisk or red biro mark) and the student must write what they are going to do about their inadequacies. All this should take about 10-15 minutes. During this time I hand back the previous week's assignment and talk to each student individually about their approaches, their untidy or tidy writing, their nomenclature, and their lack of explanation. I also mark the roll for a number of administrative and academic purposes. (The students receive 1/2 mark for each tutorial participated in.) This 10-15 minutes also gives me time to cope with late-comers.
For the next 10 minutes we have two student presentations. The students are nominated the week before and told which Practice questions they will be asked to present solutions to during this session. Practice questions are chosen because they are easier to present than Hand-in questions, and not all students will have attempted the Practice questions. Hence there will be a freshness to them. In addition, the students can concentrate on the presentation and not get lost in difficult mathematics. I mark them out of 4, which together with their participation mark gives them a possibility of 10%. I tell them how many marks they were awarded, and point out the strong and weak points of their presentation. Students do not have to memorise their presentation, but will score low if they simply write the solution on the board from a piece of paper in their hand without saying anything. Questions and comments are called for from the rest of the class after every student presentation. I sit down the back of the classroom.
When the student presentations are over I go through each Hand-in question one at a time (H1, H2, H3, etc), calling for comments on the model solutions or giving extra explanations on the more difficult parts. If I detect that a particular point has been generally misunderstood, or not understood at all, I tell the class that we will do an extra example near the end of the tutorial - and I mean we, since I get the students to start the example and tell me how they are progressing. I soon find out who can't even start the solution to the question!
If there is any time left, I give a thumb-nail sketch of how to handle particular points in the next assignment, and indicate the tricky parts to watch out for. As they are leaving, the students hand in the assignment that they have self-corrected. I do not accept assignments after this unless prior arrangements have been made.
I then take these assignments back to my office for marking. I note problems that I have with reading the scripts, for example, the difficulty of distinguishing 2 from z, S from 5, l from 1, 1 from 7, etc. I mark out all unnecessary $\Rightarrow$ and =. I particularly comment on lack of explanation in differentiation, as in
y = 3x^2+7
= 6x
- writing dy/dx heavily in red at the beginning of the second line.
Finally, I put a general pertinent comment at the beginning of each script, indicating that they seem to understand this topic or are weak in a particular area. Since the students have already self-corrected and made comments, I usually need to spend only about 5 minutes on each script. This means that since all students do not hand in scripts I have to allow about 60 to 80 minutes each week on marking for each tutorial group, which I don't think is unreasonable.
Although I read and mark the assignments with some care I do not base a major portion of the assessment process on the assignments. In fact assignments attempted count for only a total of 1 or 2 bonus marks which can be used to push students on the border of any cut-off area (fail/conceded pass, conceded pass/pass, pass/credit, credit/distinction, distinction/high distinction) into the area above. Thus a student who attempts and hands in all his/her assignments, and was on 48 will pass; while a student who hands in nothing and was on 49, remains on 49 with a conceded pass. My rationale is that the latter student has provided me with no extra information to influence his/her case.
The reason that I do not place much mark emphasis on the assignments is the classic one that unfortunately some students will copy out the sample solutions (even including any deliberate errors) from previous semesters and submit them as their own work. They then proceed to fail all exams, even though their assignment work seems to indicate that they understand the course very well.
Having a small mark value for assignments also encourages honest collaboration, whereas having a high mark is a recipe for dishonest collaboration where one student does all the work and his/her friends submit the solutions as if they were their own. Honest collaboration is a much more positively healthy approach, where the less-informed student discusses with and learns from his/her friend. Then he/she writes and submits the solution in their own words and symbols without having a copy of their friend's work nearby.
When teaching this and other courses, I always emphasise the importance of sketching curves and using diagrams. I show how to quickly sketch straight lines, absolute value functions, quadratics and rectangular hyperbolae well before calculus is introduced. I then obtain the same curves using a computer software package such as ANUgraph (for service courses) or MATHEMATICA (for maths majors). The computer sketches are investigated using the ZOOM feature, and cubics and other more difficult curves can be included. Inequalities are investigated by comparing two sketches on the same set of axes. Computer sketches must be included in the student's hand-in assignment scripts.
The other computer package that I teach students is the Equation Editor associated with Word. I introduce this about half way through the course, so that the first six assignments may be written by hand, but the final six must have some mathematical typing. I do not insist that the whole of an assignment is typed, just the first hour of the script's preparation time. In this way all my Business School mathematics students should be able to type up quantitative and symbolic material in addition to words.
There are two main sections in Krantz [1993] that are pertinent to the above discussion. These are Section 2.5 on Homework, and Section 2.14 on Problem Sessions, Review Sessions, and Help Sessions.
In summary Krantz says:
This new approach to mathematics tutorials has many advantages. The students bring some completed work to the session, receive model solutions, and learn to read mathematics and be highly critical of their own efforts. Each student is given the opportunity during their presentations to verbalise their mathematical knowledge in a particular area. All students in the class are concentrating on the same problems in the same order. Students can therefore help one another.
More time can be spent by the tutor in fine-tuning the student's written solutions. Since the students learn to self-correct, the tutor doesn't need to spend time looking for all the errors in each solution.
Such a structured approach to tutorials means that more students can be cared for in a tutorial group than in the old approach, and every student can be catered for at their own rate of learning. When a tutorial session is finished, all of the students should feel that they have learnt something, and that they have information which has improved their knowledge and understanding of the topic.
School of Information Technology
Bond University
Queensland