Poor teaching leads to the inevitable idea that the subject (mathematics) is only adapted to peculiar minds, when it is the one universal science, and the one whose ground rules are taught us almost in infancy and reappear in the motions of the universe.
~ H.J.S. Smith
The mind is not a vessel to be filled. It is a fire to be kindled. ~ Plutarch
The moving power of mathematics is not reasoning but imagination.
~ Augustus De Morgan
The Universe is a grand book which cannot be read until one first learns to comprehend the language and become familiar with the characters in which it is composed. It is written in the language of mathematics. ~ Galileo
The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver. ~ I.N. Herstein
I keep the subject (mathematics) constantly before me and wait till the first dawnings open little by little into the full light. ~ Sir Isaac Newton
Kelli (@Singapore_Math) tweeted me the “Ivy League Math Funeral [1872]” story motivated me to think more about the persistence of negative attitudes towards mathematics in societies around the world.
In my previous post “Let’s Start A Numeracy Day!”, I asked:
(1) Why is there such a focus on literacy when numeracy is just as important?
(2) What are the dangers of innumeracy?
Read also one of my old post on math education: Crank Dat Calculus
How early can you teach Math to kids?
I say: It can never be too early. You just need to keep some order. Multiplication should be taught after addition, and negative numbers after subtraction. Kids should remember multiplication by heart at the age of seven. 4-year old kids can understand negative numbers. Show geometrical shapes, for example, Platonic solids, to babies.
Now, I ask,
What is the origin of the negative attitudes towards mathematics?
And how societies have accepted this? How has this acceptance developed?
I’ve been searching on the history of teaching mathematics to look for clues to understand better why societies have adopted such attitudes and why they persist till today.
Mathematics has a long history, dating back to Mesopotamian culture over 4,000 years ago. Soon after, many other ancient cultures also began to develop their mathematics.
I’ve looked into …
… Mathematics in Mesopotamia
… “The teaching of mathematics in Ancient Greece”
… In ancient India
… In ancient China
… In the medieval Arabic-Islamic civilization.
… Mathematics Through the Middle Ages (320-1660AD)
… In North America
Watch the BBC documentary SCIENCE & ISLAM 1 to 6 | The language of science
Watch also (a contribution from @grey_matter, on Twitter) the videos about the history of mathematics, presented by Oxford professor Marcus du Sautoy at
A Story Of Maths For All
The Babylonians created a number system, sexagesimal (base-60), developed a practical geometry, elementary algebra-like calculations, and fractions. The Egyptians used a decimal system (base-10), and developed a practical geometry that enabled them to construct and determine the areas of many simple figures, most famously demonstrated by the construction of the pyramids. Greek mathematics went beyond the practical elements of other ancient traditions and looked at abstract, philosophical, and mystical aspects of numbers. In ancient India, teaching method emphasized rote learning, the 3-step process of teaching math rules: memorize, practice, and then, rationalize. (Many students didn’t reach last stage.)
In ancient China : Mathematics education in ancient China was quite developed. During the period of the Sui Dynasty (A.D. 581-618). In the Tang and Song Dynasties, the size of the mathematics school was further enlarged. Later, however, from the Ming Dynasty to the Qing Dynasty, mathematics education declined because the Imperial examination emphasized writing essays more.
In ancient Greece, Arithmetics and Geometry were considered to be separate subjects. Even within Arithmetic itself there were two forms:
(1) A practical arithmetic that was taught to the middle and artisan classes and was very much a calculation based subject.
(2) the science of numbers, was the preserve of a few of the upper classes who had the time and money for a more lengthy education, but learned the minimum of Arithmetics or Geometry.
The upper class wasn’t too interested in Arithmetic because that was used by the artisan classes.
Many Educationalists have advised against prolonged consideration of mathematical ideas since they believed that it led to too great a level of abstraction and drew the mind away from the realities of the world they lived in.
This article asks,
what were the causes of the decline of scientific work in Islam, and why did the gap in modern science and technology become so great between the West and Islam from the end of the sixteenth century?
It answers:
the rational sciences were flourishing because of the prosperity of these regions and the stability of their civilization.
The early Middle Ages of Europe (300AD to 1100AD) saw little advance made in the field of mathematics. Major advances in mathematics were a thing of the past, even the errors in ancient texts were taken as truths in mathematics of the time. Kline (1977) suggests that the reason no advances were made in the Middle Ages is because those in power were too concerned with the civitas dei and the preparation for the latter world. Whereas the reason for the stagnation of the Roman Empire was for the exact opposite reason, they were too concerned with the civitas mundi or practical results. Through this and other historical evidence it becomes apparent that mathematics cannot bloom under either of these social climates.
No major discoveries or advances were made by Byzantine mathematicians, they were too busy preserving knowledge from ancient Greece. The Church wanted to maintain its monopoly of knowledge, and felt threatened of any new knowledge.
St Augustine is accredited with saying:
“whatever knowledge man has acquired outside of Holy Writ, if it be harmful it is there condemned; if it be wholesome it is there contained”
during his time (354 to 430 AD) and seems to have forecast the attitude of the Church toward mathematics up until about 1100 AD.
William Heard Kilpatrick majored in mathematics at Mercer College in Macon, Georgia. His mathematical education included some graduate work at Johns Hopkins University, but his interests changed and he eventually attended Teachers College and joined the faculty in 1911. William Heard Kilpatrick’s major contribution was his attack on the doctrine of “formal discipline,” (Doctrine of formal discipline is the theory held years ago in education that the learning of subjects such as Latin and mathematics strengthened the individual’s reasoning powers.) an attack dating from 1901 but repeated at intervals, always with fresh evidence. Traditionally, education was supposed to exercise and develop the mental faculties, so that the educational value of geometry, for example, lay in its power to this doctrine from the start. He proceeded to test it by use of the “transfer” experiment. Kilpatrick rejected the notion that the study of mathematics contributed to mental discipline. His view was that subjects should be taught to students based on their direct practical value, or if students independently wanted to learn those subjects. Kilpatrick proposed that the study of algebra and geometry in high school be discontinued “except as an intellectual luxury.” According to Kilpatrick, mathematics is “harmful rather than helpful to the kind of thinking necessary for ordinary living.” In an address before the student body at the University of Florida, Kilpatrick lectured, “We have in the past taught algebra and geometry to too many, not too few.” Thorndike conducted a series of experiments beginning in 1901 that cast doubt on the value of mental discipline and the possibility of transfer of training from one activity to another. These findings were used to challenge the justification for teaching mathematics as a form of mental discipline and contributed to the view that any mathematics education should be for purely utilitarian purposes …. This led to the fragmentation of arithmetic and the avoidance of teaching closely related ideas too close in time, for fear of establishing incorrect bonds. According to one writer, “For good or for ill, it was Thorndike who dealt the final blow to the ‘science of arithmetic. According to David Snedden, the founder of educational sociology, and a prominent professor at Teachers College at the time, “Algebra…is a nonfunctional and nearly valueless subject for 90% of all boys and 99% of all girls–and no changes in method or content will change that.”
In Japan, the soroban, or abacus is making a comeback. Read and listen to the Story.
Sherri (@singamathexpert, on Twitter) tweeted me this article:
America’s Educational System is Doing Alright!
Consider the following story: Waiting for Superman or Godot?
For those who may not know: Waiting for Superman and in Wiki, and Waiting for Godot
Do Smaller Schools Mean Success?
Read the article : PBS NEWSHOUR reports
An advice coming from an 11-year kid may have a more powerful impact than from adult. Consider Ethan Brown, 11, of Bethel, a math whiz, offers this advice for kids who are struggling with math :
You shouldn’t think you aren’t smart if you can’t figure something out. Every time you get an answer wrong, you’re one step closer to getting it right. You’ll know not to (make the same mistake) again, and you might try another way to solve it. This will make you a better math student.
[@grey_matter (On Twitter) tweeted me the story]
Caution: The story of this boy may reinforce one of the myths and misconceptions about mathematics, namely, “To be good at math you have to be good at calculating”. I believe it is a damaging idea, this will perpetuate math anxiety that many students have about math.
Remember that being a wiz at figures is not the mark of success in mathematics. Modern mathematics is a science of ideas, not an exercise in calculation.
Another myth about math that needs to be debunked is, “Men are naturally better than women at mathematical thinking”. IT IS NOT TRUE!
There are many studies about this. One of them is the following:
Study: No gender differences in math performance
I’ve just come across the following study: children who had the greatest exposure to Jalan Sesama (an Indonesian version of Sesame Street program) improved significantly in literacy, mathematics, early cognitive skills, …
Read more ….
Charlotte Thomson Iserbyt wrote the book “The Deliberate Dumbing Down of America”
Presenting the author
Read the reviews by Amazon of “The deliberate dumbing down of america – A Chronological Paper Trail: A Chronological Paper Trail
“The destruction of America’s education system: This book argues that the academic meltdown in our public education system is intentional. It asserts that change agents have been working at the Education Department to change curriculum, not to improve teaching but to promote a socialist agenda. Their role is to create schools which will mold obedient citizens who no longer have the knowledge and skills to improve their lot in life, but are dependent on governement/multi-national companies’ guidance to survive. The system will create imprisoned citizens that will be managed from cradle to grave to serve the needs of the state’s managed economy. The book is clearly written,copiously documented, and finally asnwers the question “Why can’t our kids read, write, and count?” A must-read for anyone with children. It presents a scary view of America’s future if nothing is done to bring back our schools to the excellence of the turn of the century.”
What’s your reaction?
[To Be Continued]
More info :
(1) David Snedden
(2) How Dewey Lost: The Victory of David Snedden and Social Efficiency in the Reform of American Education
(3) The Effects of History of Mathematics on Attitudes Toward Mathematics of College Algebra Students
(4) Teaching of Mathematics in Singapore Schools
(5) Gender Discrepancies in Mathematics
(6) @sjaubert (Twitter) made the following contributions: This document and this one
(7) Official Curriculum in Mathematics in Ancient China: How did Candidates Study for the Examination?
(8) A Quest to Explain What Grades Really Mean
(9) Contributed by @WorldClassMath (Twitter) Willingham: 3 brain facts every educator should know
(10) A history of Zero
(11) Development of Mathematics in Ancient China
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