By
Dr Matthew Dean
Twenty years ago, high school graduates entering university could pass the Leo Howard Test. Now they struggle with differentiation, trigonometry, algebra and even adding fractions*. We find now, that students are aware of these topics but have little confidence or ability with them, and that first year university students are struggling to pass even watered–down courses: they may study the first year material, but they lack the mathematical foundation to be able to implement it. Let us consider the pedagogical changes which we have adopted in our high school mathematics over the last twenty years. We have
1. introduced a spiral rather than orderly sequence of topics,
2. replaced final and comprehensive assessment with frequent small assessment items, and removed emphasis from skill development,
3. introduced writing tasks into mathematics,
4. emphasized statistics, modelling and graphics calculator use and
5. changed to a complicated, criteria based assessment.
In this essay we review these changes and their effects on our students’ confidence and thinking.
* Education Seminar, Dept of Mathematics, The University of Queensland, 2009
1. An orderly syllabus
The most popular textbook throughout known history begins with simple, familiar ideas and develops carefully and consistently by sound logical reasoning toward its final conclusions. It has been the standard for 2000 years. It is Euclid’s Elements.
The reason this text has been so popular is not because people value points, lines and triangles, though these are useful concepts. Rather, the popularity and importance of Euclid’s Elements is due to the orderly quality of the exposition:
- every step is sound.
- all the parts fit together, and
- each part builds upon the next in a logical fashion, so that
- the final conclusions of the work hold true with the same certainty as the initial postulates.
In antiquity, the Greek schools valued the training of geometry so highly that they posted above their doors the phrase
Let no one come to our school who has not learned the geometry of Euclid.
For 2000 years, Euclid’s Elements has been the standard of perfection for human reasoning. The field of mathematics has developed emulating this standard. No other art or science has offered the human mind this quality of reasoning. Thus, mathematics is the backbone of the sciences, technology and engineering. Like a strong frame, proper mathematical training, such as based on Euclid, offers the human mind an ability to think clearly, consistently and with certainty. These are valuable qualities for any walk of life.
Unfortunately, the format of our high school maths syllabus and of its supporting texts, is chaotic by comparison. Rather than following a gradual and orderly development, we find ourselves jumping frequently from one topic to another unrelated topic. Working through Euclid’s Elements engenders a feeling of calmness and confidence. But our present high school courses offer constant distraction.
What kind of thinking ability will this encourage our students to develop?
2. Developing significant thinking ability
Book One of Euclid’s Elements consists of a few initial remarks followed by 48 carefully sequenced propositions. Each proposition is deduced in about five to ten deductive steps from either previous propositions or from the initial remarks. Thus this book contains long chains of deductive reasoning.
The ability to creatively think through long chains of deductions to arrive at a conclusion can be developed in good mathematics courses. This significant thinking ability does not come easily however, and lots of practice is required. Students should begin with simple problems and build gradually up to those requiring many steps to solve.
Like developing the ability to play beautiful music on a piano, much practice is necessary. The required practice is often considered to be too routine, pointless or boring. But those who attain the final goal can report that it is all worth the effort.
A good mathematical training develops the ability to carefully think through long problems. The main benefits of this are
• self confidence – assurance in one’s ability to think, and
• freedom to apply this ability independently, rather than needing to be constantly tethered to an external authority.
By adopting our present high school syllabus, we have failed to provide an opportunity for students to develop this ability to think through long chains of deductions. We have done this by
1. frequently jumping from topic to topic,
2. providing too few practice problems on the standard topics
3. removing final, comprehensive examinations, and replacing them with frequent, small assessment items
Unfortunately we have listened to the voices which regard skill development as something too routine or old-fashioned. Our result, is that a generation of Queensland students vaguely know about various mathematics topics, but lack the confidence and ability to actually work within these topics. Our result is that a generation of students lack the confidence and independence which accompanies significant thinking ability. We are now dumb, listless consumers instead of capable, thinking producers. We have not done the practice required to play beautiful music.
3. Subject Integrity
Learning to write coherently is valuable for everyone. Very few would oppose the teaching of writing in high school English. However, putting writing tasks into mathematics is as appropriate as calculating derivatives in the middle of Hamlet, or playing the tuba while running a marathon, or mixing chemicals while watching an opera. If writing is handled adequately in English, it does not need repetition in an unrelated discipline. On the other hand, if English is not teaching writing properly, then the English syllabus requires some rethinking (and rewriting).
A balanced curriculum does not focus on one discipline to the detriment of other disciplines. We now have writing tasks in mathematics , physics and chemistry while students’ mathematical ability is suffering. The supposed justification for the absurdity of putting writing into mathematics is that ‘scientists need to write’. While scientists do write, writing is not the characteristic feature of their work. The characteristic feature of science is sound thinking – an ability which is developed by doing mathematics.
Popular technology readily assists us with spelling and grammar, but it does not make our thoughts coherent or well sequenced. It does not help us think soundly through a problem to a solution. This ability is developed by doing mathematics. Thus, with the rise of technology, good mathematics courses, are as important as ever.
4. Proliferation of Topics
(a) Statistics
Statistics has been expanded in high school mathematics. Much of statistics is very difficult and will remain a mystery to students unless they first study university level analysis. Presenting mysteries for students to believe in, may have value, but does not develop confidence or thinking ability in the same way that mathematics does. On the easier, descriptive statistics, it seems to me that we are spending a lot of time in high schools learning very little.
(b) Curve Fitting
Another topic which now receives unusual emphasis in our high schools, is that of curve fitting. That is, fitting data points to lines, polynomials, log, exponential or trig functions. This is referred to as modelling, bearing a resemblance to the modelling work of physical scientists.
Curve fitting is only one of hundreds, if not thousands of numerical problem types which face scientists and engineers (see, for example, the lists of numerical algorithms in the NAG library or in Numerical Recipes). Curve fitting is not an important enough problem to receive undue emphasis. It is more important for students to develop confidence and thinking ability by becoming proficient in the established basics of geometry, algebra, trigonometry and calculus. This is also the shortest path to accurate modelling.
(c) Technology
Software and hardware come and go so quickly, that the current technologies will almost certainly not be used in another ten years. A good mathematics course however, will always be worth studying, as it not only introduces the unchanging language of the physical world, but also develops the thinking ability of ones own mind.
(d) Advanced Topics
In the last twenty years wonderful topics such as differential equations and group theory have appeared in high school mathematics. By stretching students thinly across too many topics, students end up being vaguely aware of many topics, but lack the ability and confidence to work with any of them, even the basics.
5. Complicated Assessment
A principle of modelling is Ockham’s razor. It is the principle that
the simplest explanation or strategy tends to be the best.
If we apply Ockham’s Razor to the task of assessing mathematics ability, the obvious choice is
a mark out of 100.
Percentages are immediately understandable to everyone. Any other choice of assessment system, whether in numbers or words, will be less familiar, less simple and require explanation.
Over the last twenty years, teachers, students and parents throughout Queensland have wasted a lot of time and effort struggling to understand the complicated assessment systems we have adopted. The opaqueness and lack of consistency of our present assessment system
• discourages students from studying hard to improve their performance,
• wastes very many hours of every maths teacher’s time, taking them away from helping students learn,
• excludes parents, and tutors from their appropriate roles in students’ learning and
• lends itself to corruption.
Instead of applying Ockham’s razor to mathematics assessment, we unfortunately, listened to the fashions of pedagogy and are suffering under its weight and absurdity.
Conclusion
Through studying a good mathematics course, students can develop the ability to think clearly and consistently, develop a confident and independent mind, and become familiar with the language of the physical world. A good mathematics course demands much practice, and builds in an orderly way from simple to complex problems.
Queensland high school mathematics is not such a course. Our high school graduates’ lack of confidence and ability in thinking, is attributable to the chaotic, cluttered and superficial nature of our high school syllabus. Its main obstructions to significant thinking development are the proliferation of introduced topics (including writing tasks, and overemphasis on statistics, curve fitting and certain technology), the consequent lack of emphasis on basic skill development (geometry, algebra, trigonometry and calculus), and the overly complicated and unreliable system of assessment.