Pedagogic Research in Maths for Science

1. Dr. Martin Grayson, semi retired from the Department of Chemistry, the University of Sheffield, Sheffield S3 7HF. Email m.grayson@shef.ac.uk Pedagogic Research in Maths for Science

2. The Problem • The diverse educational background of chemists – no longer Maths, Physics and Chemistry A-levels. • Chemistry, Physics and Maths are disjoint: Consequences. • The secondary subject has political problems whether it is maths or not. • New styles of teaching might be required.

3. Research • Some research has proved there is a problem, but is this welcome? • The biggest problem is the general lack of fluency in algebra. • Improvements in algebra training can alleviate the problem but I suspect they will be incremental improvements, not a change of technology which will give easy success. • I believe without proof that the average student needs up to 200 hours of work to be proficient....

4. Subject Centres • Look at MathTutor – Wavelength Volume 2 Issue 1, from the Academy Physical Sciences Centre and http://www.mathtutor.ac.uk/ • Notice few people have published their extensive data on the decline of mathematical ability in students over time. • The educational diversity of students makes it inevitable – nobody's fault. It is even a good thing?

5. Algebra as a Competence • Ability in algebra is a competence like driving a car and perhaps should be taught as such using competence based methods. • Attempts have been made to sidestep a requirement for algebraic competence by teaching spreadsheets and computer programs instead. • Would the professional societies accept that!

6. Competence Based Skills • Mental arithmetic. • Factorization. • Algebraic manipulation. • Mechanical differentiation and integration skills. • Imaginative integration skills. • Is practicing and testing these as five core skills a good teaching method?

7. Added value? • How are the students final marks in the subsidiary correlated with the choice of A-Levels? • (Experience suggests there is a big spread of final abilities, bearing a stronger correlation to the number of workshops attended but equally some comparatively poor attenders scored very highly because the essentially school level work was well within their abilities.)

8. Self Study • Computer aided? Computers can collect our data for us without it seeming that the students are being used as guinea pigs! • I believe the first workshop on a new skill is of paramount importance. (Can we prove this without having an unethical control group?) (My explanation for this is a bit new-agey but it might be right. • Is it possible to collect data on the acquisition of new skills?

9. Peer Instruction • Some students find maths intimidating. • Peer instruction can help but research has shown in requires marked preparatory work before each workshop to be effective. • One experiment with streaming was very unsuccessful. If the weaker students are taught together they must have more time allocated, (politically how?), otherwise as a group they fall behind.

10. Can we Face Maths at 9:00am • I am pretty sure that new material is best introduced in the morning when the brain is fresh and has the rest of the day to mull over..... • Is this practical except for self study. Universities encourage us to use all the times in the timetable to allow efficient room usage. Then the students are encouraged to give feedback about the timetabling of their studies and wow: 9:00 and 17:00 are not popular! • Everyone wants to teach and learn at 11.

11. Computers? • A computerized learning method such as WebCT or one of the programs being developed at the subject centres can surreptitiously get time of day data. • It can also produce data on numbers of failed attempts leading to an automatic problem difficulty grading. • This leads us too soon to my final point in the abstract.

12. Expert Error • We think we know what is easy and what is difficult. Students think they also know this and come up with different lists. • Get the staff and students to separately grade questions by difficulty. It may produce some surprising results. • Some problems, imaginative integration and quantity estimation are unclear how to successfully do. How do experts do them?

13. An Aside on Expert Error • The classic expert error occurs when an expert has a good and probably complicated paradigm which is not applicable to the problem at the current time but he applies it and has all the mental apparatus from his system to convince everyone that he is right. • It is not quite the same as mark Twain's aphorism: When the only tool you have is a hammer, all your problems look like nails.

14. Learning Styles • At schools now there is often an emphasis on different learning styles for different pupils. • If possible organized workshops can include peer instruction. If there is CAL these workshops might be optional but only for high scorers in the CAL tests? • Problems should be more problem like. Not just plugging in different numbers to examples already given.

15. Student Attitudes • Student attitudes to secondary subjects such as Maths for Chemists and Chemistry for Engineers can be hard-nosed and quite negative. • To counter this we need to first find out what the attitudes are. • There is probably always going to be an aspect that some of the things you have to be competent in to practice a profession are not a bundle of fun.

16. Estimation and Approximation • Ridiculous answers are frequently given for problems without comment that this number cannot be right. • In the medical field people have even been killed because of calculation or solution making mistakes which were wrong by orders of magnitude. • Before we can teach Estimation we must first find out how we do it. Most courses assume students learn it magically by osmosis from their teachers.

17. Approximation • Significant figures – values have the significant figures which are displayed by the calculator. • Dinosaur story.......... • Estimation and significance need more exposure in a generic skills module.

18. The Algebra Problem Again • To effectively teach algebra we need to know how the different methods of competently doing algebra work. This might illuminate the difficulties. • Neuro researchers seem to have proved algebra and arithmetic are potentially disjoint skills and algebra is related more to language than to arithmetic. • Do student marks for writing and algebra show this correlation?

19. Summary • I hope I have shown that it would be very useful to have more knowledge of both how the expert performs the various mathematical competences and the details of how the students learn them, particularly in the area of problem solving. • Here is my list of important questions to be answered:

20. Open Questions • How many hours of work does it take to acquire basic algebraic competence? • Do experts and novices do algebra in the same way? • Does mathematics actually matter in professional practice? (Some horrible thoughts) • Is CAL more effective than pencil and paper? • How do we do estimation? Can it me taught?

21. More Questions • Are our problems challenging enough as real problems? • Should you learn your algebra when you are young? ....... What are the implications for adult learners and will research into this be ageism? • Is a depth of understanding as important as mechanical competence? (What does that squiggly line mean?)