Adequacy of pre-university mathematics
curriculum as a preparation for chemistry
undergraduate studies
Joseph N. Grima & Alfred J. Vella
Department of Chemistry, Faculty of Science,
University of Malta, Msida MSD 06, MALTA
E-mail: joseph.grima@um.edu.mt
Tel: (+356) 2340 2274
WWW: http://staff.um.edu.mt/jgri1
Malta & its University
Malta: Size: 126 square miles, Population: 380,000
The University: Traces its origin to the founding of
the Collegium Melitense in 1592.
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Science & Mathematics:
“Without mathematics the
sciences cannot be
understood, nor made clear,
nor taught, nor learned.”
 “Mathematics is the door and
key to the sciences”
Roger Bacon (1214-1292)
… chemistry is no exception!
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Chemistry … (1) Is math really required?
Chemistry is no exception … Math is required!
A non specialist looking at some physical chemistry textbooks
may think these are textbooks in advanced mathematics!
Atkins & de Paula, Atkins' Physical Chemistry 7th edition , Ch. 23
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Chemistry … (1) Is math really required?
‘A facility in mathematics is an essential
part of the armoury for all chemists.’
Mathematics in Chemistry Degree Courses –
The Royal Society of Chemistry, UK, Sept. 1996
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Chemistry … (2) What math is required?
A chemist’s minimum math skills must include skills in:
1. The handling of indices and logarithms;
2. Equations functions and graphs;
3. Differentiation and integration (including partial
differentiation and differential equations);
4. Basic trigonometry (including trigonometric identities
and polar coordinates);
5. Statistics, regression analysis and error calculations;
6. Vectors; etc..
Mathematics in Chemistry Degree Courses –
The Royal Society of Chemistry, UK, Sept. 1996
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Chemistry … (2) What math is required? …
Topics range: From ones normally covered in
some 16+ syllabi (e.g. indices) to others that are
usually part of mathematics degree courses (e.g.
partial differentiation).
Required proficiency: Students must have
grasped the math to the extent that they can use it
with confidence in their chemistry courses.
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Chemistry … (3) When should math training be
given? Ideally before and during degree course.
1
Many universities include math courses as part of
their ‘chemistry’ curriculum.
 University of Exeter (UK), University of Bologna (Italy):
About 25% of their first year.
 University of Perugia (Italy): About 25% of their first two years.
2
Importance of an adequate pre-university math
education is often reflected in the entry
requirements for chemistry degree courses.
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Entry Requirements for Chemistry Degree
Courses …
University of London, Imperial College, UK
‘Candidates should normally offer three A levels
including chemistry and mathematics.’
University of Cambridge, UK
‘If you plan to take chemistry through to the
second year and beyond, we recommend strongly
that you have A level maths (or an equivalent) —
AS is not sufficient.’
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ASIDE: What is an A-Level Exam?
A chemistry A-Level syllabus is similar to the syllabus in
a US first-year BS course with some additional organic
chemistry.
Typical question: UoM June 2003 Chemistry A-Level, Paper II, Q6
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Scope of paper:
1. To investigate how well the education
system in Malta is preparing its pupils for
degrees in chemistry by providing them
with the appropriate mathematical skills;
2. To make a few suggestions on how this
important minority can be provided
with the required mathematical skills.
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Chemistry Degrees in Malta:
‘Joint Honors’ BSc(Hons) / Science BEd(Hons)
Students choose two principal subject areas from:
Biology – Chemistry – Computer Science – Informatics –
Mathematics –Statistics & Operations Research – Physics
Common combinations involving chemistry
include:  Chemistry & Biology
Favorite combination
 Chemistry & Physics
No ‘single honors’ Chemistry program offered!
http://home.um.edu.mt/science/
http://www.educ.um.edu.mt
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‘Joint Honors’ Chemistry Degrees
… The problems
1. Teaching time is severely limited … hardly enough
time for all ‘core chemistry’ … Finding enough time
for Math classes at university is extremely difficult!
2. Impossible to impose any stringent mathematical
entry requirements:
• Second subject taken as ‘major’ will
inevitably require a different
preparation;
• No. of students majoring in chemistry
is small
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The math education of Malta’s chemistry
students in ‘Biology & Chemistry’ stream
Pre-university math
education
The math education of
chemistry students at
university
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Education System in Malta
11+ exam
Primary
Age:
|
|
5
|
|
6
|
|
7
|
|
8
16+ exam 18+ exam
Secondary
6th
Form
University
| | | | | | | | | | | | | |
| | | | | | | | | | | | | |
9 10 11 12 13 14 15 16 17 18 19 20 21 22
General Science
Chemistry
Mathematics = Compulsory
?
Math (optional)
http://www.um.edu.mt/courses/
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Pre-University Math: Ages 16-18: The 18+ exam
Mathematics NOT taught to all students.
 At age 18, students take the 18+ ‘UoM Matriculation
Certificate’:
1. Two subjects are studied at Advanced Level **
2. Three subjects are studied at Intermediate Level
(~ 1/3 of that of an Advanced Level) **
3. Systems of Knowledge (General Studies)
** Must include at least one from each of these three groups:
Group 1: The Languages
Group 2: Accounts / Economics / Marketing / Computing & IT
Group 3: The Sciences
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Pre-University Math: Ages 5-16: The 16+ exam
Mathematics taught to ALL students.
… but there are still some problems!
Problem: The differentiated examining mode (to make these
exams more accessible to students with different strengths
and abilities).
It is possible to pass the mathematics 16+ examination
through a less demanding examination paper based on a
somewhat curtailed syllabus (Paper B)
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Practical Implications: The 18+ Exam
for students in the ‘Biology & Chemistry’ stream
Group 1: The Languages:
Students may freely choose 1 subject from this group at I-Level
Group 2: Accounts / Economics / Marketing / Computing & IT:
Students may freely choose 1 subject from this group at I-Level
Group 3: The Sciences:
Students must choose:
1. Chemistry & Biology at A-Level (the main subjects)
2. Physics at I-Level (reqd. for Physical Chemistry)
… This leaves no space for mathematics!
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The ‘Biology & Chemistry’ Stream –
Problem # 1a:
The mathematical skills of most students majoring
in chemistry are very limited … just a pass in the 16+
exam
Problem # 1b:
Some students majoring in chemistry may have
opted for the less demanding 16+ examination math
paper based on a somewhat curtailed ‘Paper B’
syllabus as opposed to the more traditional and
complete ‘Paper A’ syllabus.
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Problem #1b: The ‘Paper B’ problem
% of Pupils who sat for Paper B
There seems to be a trend for more students
to opt for the easier ‘Paper B’:
25
22.2%
20
15
10.5%
10
5
0
0.0%
0.0%
1996
1997
1998
Year of examination
1999 = current
1st years
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Why worry?
% who sat for Paper B
Problem #1b:
The ‘Paper B’ problem
25
22.2%
20
15
10.5%
10
5
0
0.0%
1996
0.0%
1997
1998
Year of examination
1999
1. Chemistry students may have then felt that their
mathematical knowledge was not of high
enough standard, (or students were ‘advised’ to
take the ‘surer path’ to a Grade 4);
2. Students are not being made aware that
chemistry and mathematics are inter-related
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Problem #2: When do students become aware
that chemistry and mathematics are interrelated? 40
34%
35
30
%
25
20
28%
21%
17%
15
10
5
0
Before
Form 3
Between
Forms 3-5
At 6th Form At University
Too late ?
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Problem #3:
No mathematics for the two years
preceding commencement of their
university course!
Result: Most mathematical knowledge is
difficult to retrieve from long-term memory.
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Problem #3: No math for 2 years …
The result:
Average % mark in ‘experiment’
The experiment: 1st and 2nd year chemistry undergraduates
students were asked to work a set of math questions taken
from 16+ past-papers
70
60
59%
60%
61%
50
37%
40
30
20
10
0
1
2
3
4
Math Grade in the 16+ exam
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Problem #3: No math for 2 years …
The equal sides of an isosceles triangle are (3x-2y) cm and
2(x+y) cm. The third side is of length 3(x-3y) cm and the
perimeter of the triangle is 46cm.
(a) Find the values of x and y.
(b) Calculate the length of each side.
Beyond the capability of most students !
… Probable cause: mathematical knowledge is difficult to
retrieve from long-term memory.
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Math education of chemistry undergraduates
Up to year 2000-2001:
Course followed: MA001 – Elementary calculus
When: First year, first semester
Content: General (not specifically designed for chemists)
Syllabus: Cartesian coordinates, equations of lines and curves,
differentiation (including partial differentiation) and integration.
No. of Hours: 1 hour per week x 14 weeks
(2 ECTS credits = 5% of First year teaching)
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Math education of chemistry undergraduates
(up to 2000/2001) …
Problem: With MA001 alone, students could not
cope with the math contained in the core
chemistry courses.
Reason: (Main) Students’ inability to do very
basic algebra, such as, simplifying equations,
working with indices and logarithms, factorizing,
etc. (These topics are usually taught at school!)
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Math education of chemistry undergraduates
(up to 2000/2001) …
Possible causes:
1. Changes in 16+ math syllabus: logs no longer
included, ‘index laws’ covered only in Paper A, etc.;
2. Time:
• MA001 teaching time was too little;
• Not enough time to assimilate the material (MA001
was taught over a single semester.);
3. Problems in transferring the ‘pure’ math learnt in
MA001 to the chemistry applications.
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Math education of chemistry undergraduates
(From to 2001/2002)
The ‘solution’: CH033 – Mathematics for Chemists*
1. Designed specifically for chemists
2. Taught through examples that chemistry students
could relate to;
3. It was made sure that no math knowledge which the
students did not have was being pre-assumed;
4. CH033 had more time and credits allocated to it:
• 4 ECTS credits = 10% of 1st Year
• Run over two semesters
• Given ample teaching time (one hour of lecture
+ two hours of tutorials per week, for 28 weeks)
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Math education of chemistry undergraduates
(From to 2001/2002)
CH033 – Mathematics for Chemists
Does it work?
Test: Three months into the course, students were asked to
sit for a ‘closed book’ test
Result: Average mark was just 43%
WHY????
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Math education of chemistry undergraduates
(From to 2001/2002)
CH033 – Mathematics for Chemists
Why students thought they did so badly in the test:
Attitude to the subject
Lecturer's fault
Inadequate pre-university Math
Time
Applying math to chemistry
Too Difficult
0
5
10
15
20
25
30
% of students
35
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Math education of chemistry undergraduates
… The problem of time
In a joint-honors degree, the number of credits
that may be allocated to ‘math for chemists’ is
necessarily limited:
25 % of year in a full ‘single honors’ program =
more that 50% of total chemistry credits in a
‘Joint honors’ program!!!
The math problem should be tackled before
students reach the university.
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But … Ensuring an adequate math education
from schools = Difficult task:
1. Different students’ strengths and abilities;
2. Different future career pathways.
– Only a moderate percentage of students will
eventually make it into tertiary education
– Different university disciplines require different
levels of expertise in literacy and numeracy
– No. of students taking chemistry at Uni is
negligible!!!
3. School time-table is also full !
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Making the necessary changes
The only solution is to optimize and strengthen
the math education at all levels:
(1) School teachers should highlight the link between
chemistry and mathematics and encourage future
chemists to take math seriously.
(2) The introduction of a new subject at secondary
school level (“Further Mathematics”, taught in
parallel to the current math syllabus).
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Making the necessary changes
(3) Sixth Forms start requiring a Grade 3 or better in
math 16+ from students taking chemistry at ALevel. (or, to oblige those students with a grade 4
or 5 to take supplementary lessons in math).
This will send the right signals to the
students in secondary schools!!!
(4) Make changes in the 18+ exam system:
Is it wise to allow students aspiring to enter
university to halt their mathematics education
(and probably also that of Proficiency in English
and Maltese) at the age of 16?
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Conclusions:
1. There are several inadequacies in the
mathematical preparation at secondary and postsecondary level of education of those students
who wish to further their studies in sciences.
2. However, there are also a number of possible
solutions which could address the situation
without causing major disturbance. Also, the time
is right to make an effort to rectifying this
particular problem which otherwise may actually
worsen rather than improve in future.
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Acknowledgments
The authors would like to thank all
chemistry undergraduates, especially
the students of the study unit CH033 for
their help in this paper.
Thank you!
Joseph N. Grima & Alfred J. Vella
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