Introduction to Mathematical Thinking
Keith Devlin
Learn how to think the way mathematicians do - a powerful cognitive process developed over thousands of years.Announcements
Course goals and syllabus
The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand years.
Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.
The primary audience is first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. If that is you, you will need mathematical thinking to succeed in your major. Because mathematical thinking is a valuable life skill, however, anyone over the age of 17 could benefit from taking the course.
The course starts on Monday September 17 and lasts for seven weeks, five weeks of lectures (two a week) followed by two weeks of monitored discussion and group work, including an open book final exam to be completed in week 6 and graded by a calibrated peer review system in week 7.
Lecture videos are released at 10:00AM PST on Mondays and Fridays. (Week 1 is slightly different: The Monday video release (listed as Lecture 0) is a short welcome and course description, the Wednesday and Friday releases are of lectures 1 and 2.)
The lecture topics are (in addition to the initial Instructor’s welcome on the first day of class):
Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.
The primary audience is first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. If that is you, you will need mathematical thinking to succeed in your major. Because mathematical thinking is a valuable life skill, however, anyone over the age of 17 could benefit from taking the course.
The course starts on Monday September 17 and lasts for seven weeks, five weeks of lectures (two a week) followed by two weeks of monitored discussion and group work, including an open book final exam to be completed in week 6 and graded by a calibrated peer review system in week 7.
Lecture videos are released at 10:00AM PST on Mondays and Fridays. (Week 1 is slightly different: The Monday video release (listed as Lecture 0) is a short welcome and course description, the Wednesday and Friday releases are of lectures 1 and 2.)
The lecture topics are (in addition to the initial Instructor’s welcome on the first day of class):
- Introductory material
- Analysis of language – the logical combinators
- Analysis of language – implication
- Analysis of language – equivalence
- Analysis of language – quantifiers
- Working with quantifiers
- Proofs
- Proofs involving quantifiers
- Elements of number theory
- Beginning real analysis
Created Mon 13 Aug 2012 4:07:26 AM PDT
Last Modified Tue 14 Aug 2012 6:13:54 PM PDT
Last Modified Tue 14 Aug 2012 6:13:54 PM PDT
Course structure
Basic elements of the course
- Lectures – videos presented by the instructor.
- In-lecture quizzes – simple multiple-choice questions that stop the lecture, designed to assist you in pacing and monitoring your progress.
- Assignment sheets – downloadable PDF files to work through in your own time at your own pace, ideally in collaboration with other students. Not graded.
- Problem sets – in-depth problems like those on the assignment, but with a deadline for submitting your answers (in a multiple choice format). Graded.
- Tutorial sessions – the instructor distributes (PDF) or explains (video) answers to some of the previous week’s assignment problems.
- Reading assignments – downloadable PDFs files providing important background information.
- Final exam – a downloadable PDF file that you will have one week to complete before participating in a peer review process. Required to be eligible for a grade of completion with distinction.
Lectures
The video presented lectures vary in length from about 25 to 50 minutes if played straight through, but completing the embedded progress quizzes will extend the total duration by a few minutes, and you will likely want to stop the playback several times for reflection, and sometimes you will want to repeat a section, perhaps more than once. So you can expect to spend between one hour and ninety minutes going through each lecture, occasionally perhaps more.In-lecture quizzes
Each lecture is broken up by short multiple-choice “progress quizzes”. These in-lecture quizzes are essentially punctuation, providing a means for you to check that you are sufficiently engaged with the material.Slightly modified versions of the quizzes will also be released as standalones at the same time as the lecture goes live, so if you do not have a good broadband connection and have to download the lecture videos to watch offline, you can still take the quizzes. In which case, you should do so as close in time to viewing the lecture as possible, to ensure gaining maximim benefit from the quizzes in monitoring your progress. The standalone quizzes are grouped according to lecture.
Completion of all the quizzes is a requirement (along with watching all the lectures) for official completion of the course, but we do not record your quiz scores, so quiz performance does not directly affect your final grade. If you complete the quizzes while watching the lecture (the strongly preferred method, as it helps you monitor your progress in mastering the material), you do not need to complete the standalone versions.
Course assignments
An assignment will be released at the end of each lecture, as a downloadable PDF file. The assignment is intended to guide understanding of what has been learned. Worked solutions to problems from the two weekly assignments will be demonstrated (video) or distributed (PDFs) in a tutorial session released the following Wednesday (so in weeks 2 through 6). The tutorial sessions will be released at 10:00AM PST.Working on these assignment problems forms the heart of the learning process in this course. You are expected to form or join a study group, discuss the assignment problems with others in the group, and share your work with them. You are also strongly urged to assess one another’s answers. A structured form of peer review will be used for the final exam, when you will be graded by, and grade the work of, other students, randomly (and blindly) assigned, so it will help to familiarize yourself beforehand with the process of examining the work of others and providing (constructive) feedback.
Problem sets
At the end of each week, following the Friday lecture, a for-credit Problem Set will be posted, with submission due the following Tuesday. The scores on these problem sets will count toward the course grade. Though the Problem Set has a multiple-choice quiz format, these questions are not the kind you can answer on the spot (unlike most of the in-lecture quizzes). You will need to spend some time working on them before entering your answers.Though you are strongly encouraged to work with others on understanding the lecture material and attempting the regular assignments, the intention is that you work alone on the Problem Sets, which are designed to give you and us feedback on how you are progressing.
Final exam
Though the lectures end after week 5 (apart from a tutorial on the final assignment), the final two weeks are intended to be highly active ones for any students seeking a grade of distinction, with considerable activity online in the various forums and discussion groups. This is when you are supposed to help one another make sense of everything.At the start of week 6, an open-book exam will be released, to be completed by the end of the week. Completed exams will have to be uploaded as either images (or scanned PDFs) though students sufficiently familiar with TeX have an option of keyboard entry on the site. The exam will be graded during week 7 by a calibrated peer review system. The exam will be based on material covered in the entire course.
As with the weekly Problem Sets, the intention is that you work alone in completing the final exam.
NOTE: The process of peer reviewing the work of others (throughout the course, not just in the final exam) is intended to be a significant part of the learning experience and participating in the formal peer review procedure for the final exam is a requirement for getting a grade of distinction. In principle, it is during week 7 that stronger students will make cognitive breakthroughs. (Many of today’s professors really started to understand mathematics when, as graduate student TAs, they first helped others learn it!)
Course completion and final grade
There are two final course grades: “completion” and “completion with distinction”. Completion requires viewing all the lectures and completing all the (in-lecture) quizzes and the weekly problem sets. Distinction depends on the scores in the problem sets and the result of the final exam.Pacing
The pacing of the lecture releases is designed to help you maintain a steady pace. At high school, you probably learned that success in mathematics comes from working quickly (and alone) and getting to the right answer as efficiently as possible. This course is about learning to think a certain way – the focus is on the process not the product. You will need time to understand and assimilate new ideas. Particularly if you were a whiz at high-school math, you will need to slow down, and to learn to think and reflect (and ideally discuss with others) before jumping in and doing. A steady pace involving some period of time each day is far better than an all-nighter just before a quiz is due. Created Mon 13 Aug 2012 4:17:31 AM PDT
Last Modified Thu 6 Sep 2012 9:25:13 PM PDT
Last Modified Thu 6 Sep 2012 9:25:13 PM PDT
Grading & Certificates of Completion
Certificate of Completion
To receive a Certificate of Completion, you have to view all the lectures, complete all the in-lecture quizzes, and complete the problem sets with an adequate aggregate grade, as logged by the system. Determination of what constitutes an adequate grade for the problem sets will be made by the instructor and the TA’s, and will depend on the overall performance of the class as a whole. The intention is that at least 80% of the students who stay with the course to the end will receive a certificate of completion. Completion does not require taking the exam and participating in the peer review process.Completion with Distinction
Distinction depends on achieving sufficiently high scores in the problem sets and the final exam. Determination of what constitutes sufficiently high scores will be made by the instructor and the TA’s, and will depend on the overall performance of those who complete all course requirements, including the final exam and the peer review process. The intention is that about 20% of students who receive a Certificate of Completion will receive a designation of Distinction. (Distinction is thus meant to indicate what the word suggests.)Though you are strongly encouraged to work together on understanding the course content and attempting the regular assignments, you should work alone on the weekly Problem Sets and on the final exam, as they are intended to measure your individual performance.
Or simply enjoy the ride
If you find you don't have the time to do the quizzes, assignments, or the problem sets, you're still more than welcome to just stay and watch the videos!Grading policy
Grade points are awarded for the weekly Problem Sets and for the final exam. All questions on both the problem sets and the exam indicate the number of points available for that question. Problem Sets are due in at noon PST on the Tuesday after they are released. If you miss this deadline, the system will accept your solution up until 10:00AM PST the following day (Wednesday), with a 10% late penalty. Since Problem Set questions may be discussed in the Wednesday tutorial session, released at 10:00AM, the system will not accept submissions after that second deadline. Sorry, with an automated system tracking many thousands of students, we cannot offer the flexibility we'd like to. There is just one deadline for the final exam (stated on the exam sheet), with no possibility of accepting late submissions. Created Mon 13 Aug 2012 4:01:30 PM PDT
Last Modified Sat 8 Sep 2012 8:10:42 PM PDT
Last Modified Sat 8 Sep 2012 8:10:42 PM PDT
Course textbook
Completion of this course does not require a textbook. Many students prefer to have a textbook on hand, however, and accordingly I have written one specifically aligned to the course. It is a short book (just 102 pages), called Introduction to Mathematical Thinking, and is available from Amazon at $9.99 (with equivalent prices in other parts of the world). (I intended to bring out a Kindle version, but encountered problems that could not be resolved in time for this course. Sorry.) The general term for a course such as this is "mathematics transition course", and there are many books with titles that contain phrases such as "transition to advanced mathematics" and almost all of them could be used successfully with this course. Most contain much more material than my book, but they are all fairly expensive (in some cases, very expensive), which is why I wrote my own smaller, but far cheaper book.
Created Mon 13 Aug 2012 6:45:00 AM PDT
Last Modified Thu 6 Sep 2012 9:31:59 PM PDT
Last Modified Thu 6 Sep 2012 9:31:59 PM PDT
Community TAs
A designation of "Community TA” (Community Teaching Assistant) will be assigned to individuals who, in the initial course questionnaire, declare that they are familiar with the course contents or else have direct access to someone who is (such as being a current students in a similar class at a physical college or university). Community TAs will be so designated in their on-line identity, so others will recognize them. Look out for that designation on the discussion forums. We ask the Community TAs to monitor the online forum discussions from time to time and jump in if they see an individual or group who has got something wrong or otherwise needs help.
Created Mon 13 Aug 2012 3:38:59 PM PDT
Last Modified Sat 15 Sep 2012 12:29:50 AM PDT
Last Modified Sat 15 Sep 2012 12:29:50 AM PDT
Readings
1. Background Reading[7 page PDF] Assigned in Lecture 0.
2. Supplement (Set Theory) [5 page PDF] Optional reading to cover assumed knowledge of set theory.
2. Supplement (Set Theory) [5 page PDF] Optional reading to cover assumed knowledge of set theory.
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) BACKGROUND READING 1
What is mathematics?
For all the time schools devote to the teaching of mathematics, very little (if any) is spent trying to
convey just what the subject is about. Instead, the focus is on learning and applying various procedures
to solve math problems. That's a bit like explaining soccer by saying it is executing a series of maneuvers
to get the ball into the goal. Both accurately describe various key features, but they miss the \what?"
and the \why?" of the big picture.
Given the demands of the curriculum, I can understand how this happens, but I think it is a mistake.
Particularly in today's world, a general understanding of the nature, extent, power, and limitations of
mathematics is valuable to any citizen. Over the years, I've met many people who graduated with degrees
in such mathematically rich subjects as engineering, physics, computer science, and even mathematics
itself, who have told me that they went through their entire school and college-level education without ever
gaining a good overview of what constitutes modern mathematics. Only later in life do they sometimes
catch a glimpse of the true nature of the subject and come to appreciate the extent of its pervasive role
in modern life.
1 More than arithmetic
Most of the mathematics used in present-day science and engineering is no more than three- or four-
hundred years old, much of it less than a century old. Yet the typical high school curriculum comprises
mathematics at least three-hundred years old|some of it over two-thousand years old!
Now, there is nothing wrong with teaching something so old. As the saying goes, if it ain't broke,
don't x it. The algebra that the Arabic speaking traders developed in the eighth and ninth centuries (the
word comes from the Arabic term al-jabr ) to increase e ciency in their business transactions remains as
useful and important today as it did then, even though today we may now implement it in a spreadsheet
macro rather than by medieval nger calculation.
But time moves on and society advances. In the process, the need for new mathematics arises and,
in due course, is met. Education needs to keep pace.
Mathematics is believed to have begun with the invention of numbers and arithmetic around ten
thousand years ago, in order to give the world money. (Yes, it seems it began with money!)
Over the ensuing centuries, the ancient Egyptians and Babylonians expanded the subject to include
geometry and trigonometry.1 In those civilizations, mathematics was largely utilitarian, and very much
of a \cookbook" variety. (\Do such and such to a number or a geometric gure and you will get the
answer.")
The period from around 500bce to 300ce was the era of Greek mathematics. The mathematicians
of ancient Greece had a particularly high regard for geometry. Indeed, they regarded numbers in a
geometric fashion, as measurements of length, and when they discovered that there were lengths to which
their numbers did not correspond (the discovery of irrational lengths), their study of number largely came
to a halt.2
In fact, it was the Greeks who made mathematics into an area of study, not merely a collection of
techniques for measuring, counting, and accounting. Around 500bce, Thales of Miletus (now part of
1Other civilizations also developed mathematics; for example the Chinese and the Japanese. But the mathematics of
those cultures does not appear to have had a direct in
uence on the development of modern western mathematics, so in
this book I will ignore them.
2There is an oft repeated story that the young Greek mathematician who made this discovery was taken out to sea
and drowned, lest the awful news of what he had stumbled upon should leak out. As far as I know, there is no evidence
whatsoever to support this fanciful tale. Pity, since it's a great story.
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) BACKGROUND READING 2
Turkey) introduced the idea that the precisely stated assertions of mathematics could be logically proved
by formal arguments. This innovation marked the birth of the theorem, now the bedrock of mathematics.
For the Greeks, this approach culminated in the publication of Euclid's Elements, reputedly the most
widely circulated book of all time after the Bible.3
By and large, school mathematics is based on all the developments I listed above, together with just two
further advances, both from the seventeenth century: calculus and probability theory. Virtually nothing
from the last three hundred years has found its way into the classroom. Yet most of the mathematics
used in today's world was developed in the last two hundred years!
As a result, anyone whose view of mathematics is con ned to what is typically taught in schools is
unlikely to appreciate that research in mathematics is a thriving, worldwide activity, or to accept that
mathematics permeates, often to a considerable extent, most walks of present-day life and society. For
example, they are unlikely to know which organization in the United States employs the greatest number
of Ph.D.s in mathematics. (The answer is almost certainly the National Security Agency, though the
exact number is an o cial secret. Most of those mathematicians work on code breaking, to enable the
agency to read encrypted messages that are intercepted by monitoring systems|at least, that is what is
generally assumed, though again the Agency won't say.)
The explosion of mathematical activity that has taken place over the past hundred years or so in
particular has been dramatic. At the start of the twentieth century, mathematics could reasonably be
regarded as consisting of about twelve distinct subjects: arithmetic, geometry, calculus, and several more.
Today, between sixty and seventy distinct categories would be a reasonable gure. Some subjects, like
algebra or topology, have split into various sub elds; others, such as complexity theory or dynamical
systems theory, are completely new areas of study.
The dramatic growth in mathematics led in the 1980s to the emergence of a new de nition of mathe-
matics as the science of patterns. According to this description, the mathematician identi es and analyzes
abstract patterns|numerical patterns, patterns of shape, patterns of motion, patterns of behavior, vot-
ing patterns in a population, patterns of repeating chance events, and so on. Those patterns can be
either real or imagined, visual or mental, static or dynamic, qualitative or quantitative, utilitarian or
recreational. They can arise from the world around us, from the pursuit of science, or from the inner
workings of the human mind. Di erent kinds of patterns give rise to di erent branches of mathematics.
For example:
Arithmetic and number theory study the patterns of number and counting.
Geometry studies the patterns of shape.
Calculus allows us to handle patterns of motion.
Logic studies patterns of reasoning.
Probability theory deals with patterns of chance.
Topology studies patterns of closeness and position.
Fractal geometry studies the self-similarity found in the natural world.
2 Mathematical notation
One aspect of modern mathematics that is obvious to even the casual observer is the use of abstract
notations: algebraic expressions, complicated-looking formulas, and geometric diagrams. The mathe-
maticians' reliance on abstract notation is a re
ection of the abstract nature of the patterns they study.
Di erent aspects of reality require di erent forms of description. For example, the most appropriate
way to study the lay of the land or to describe to someone how to nd their way around a strange town
is to draw a map. Text is far less appropriate. Analogously, annotated line drawings (blueprints) are the
3Given today's mass market paperbacks, the de nition of \widely circulated" presumably has to incorporate the number
of years the book has been in circulation.
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) BACKGROUND READING 3
appropriate way to specify the construction of a building. And musical notation is the most appropriate
way to represent music on paper.
In the case of various kinds of abstract, formal patterns and abstract structures, the most appropriate
means of description and analysis is mathematics, using mathematical notations, concepts, and proce-
dures. For instance, the symbolic notation of algebra is the most appropriate means of describing and
analyzing general behavioral properties of addition and multiplication.
For example, the commutative law for addition could be written in English as:
When two numbers are added, their order is not important.
However, it is usually written in the symbolic form
m + n = n + m
Such is the complexity and the degree of abstraction of the majority of mathematical patterns, that to
use anything other than symbolic notation would be prohibitively cumbersome. And so the development
of mathematics has involved a steady increase in the use of abstract notations.
Though the introduction of symbolic mathematics in its modern form is generally credited to the
French mathematician Fran coise Vi ete in the sixteenth century, the earliest appearance of algebraic
notation seems to have been in the work of Diophantus, who lived in Alexandria some time around
250ce. His thirteen volume treatise Arithmetica (only six volumes have survived) is generally regarded
as the rst algebra textbook. In particular, Diophantus used special symbols to denote the unknown in
an equation and to denote powers of the unknown, and he had symbols for subtraction and for equality.
These days, mathematics books tend to be awash with symbols, but mathematical notation no more
is mathematics than musical notation is music. A page of sheet music represents a piece of music; the
music itself is what you get when the notes on the page are sung or performed on a musical instrument. It
is in its performance that the music comes alive and becomes part of our experience; the music exists not
on the printed page but in our minds. The same is true for mathematics; the symbols on a page are just a
representation of the mathematics. When read by a competent performer (in this case, someone trained
in mathematics), the symbols on the printed page come alive|the mathematics lives and breathes in the
mind of the reader like some abstract symphony.
To repeat, the reason for the abstract notation is the abstract nature of the patterns that mathematics
helps us identify and study. For example, mathematics is essential to our understanding the invisible
patterns of the universe. In 1623, Galileo wrote,
The great book of nature can be read only by those who know the language in which it was written.
And this language is mathematics.4
In fact, physics can be accurately described as the universe seen through the lens of mathematics.
To take just one example, as a result of applying mathematics to formulate and understand the laws
of physics, we now have air travel. When a jet aircraft
ies overhead, you can't see anything holding it
up. Only with mathematics can we \see" the invisible forces that keep it aloft. In this case, those forces
were identi ed by Isaac Newton in the seventeenth century, who also developed the mathematics required
to study them, though several centuries were to pass before technology had developed to a point where
we could actually use Newton's mathematics (enhanced by a lot of additional mathematics developed in
the interim) to build airplanes. This is just one of many illustrations of one of my favorite memes for
describing what mathematics does: mathematics makes the invisible visible.
3 Modern college-level mathematics
With that brief overview of the historical development of mathematics under our belts, I can start to
explain how modern college math came to di er fundamentally from the math taught in school.
Up to about 150 years ago, although mathematicians had long ago expanded the realm of objects
they studied beyond numbers (and algebraic symbols for numbers), they still regarded mathematics as
4The Assayer. This is an oft repeated paraphrase of his actual words.
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) BACKGROUND READING 4
primarily about calculation. That is, pro ciency at mathematics essentially meant being able to carry out
calculations or manipulate symbolic expressions to solve problems. By and large, high school mathematics
is still very much based on that earlier tradition.
But during the nineteenth century, as mathematicians tackled problems of ever greater complexity,
they began to discover that their intuitions were sometimes inadequate to guide their work. Counter-
intuitive (and occasionally paradoxical) results made them realize that some of the methods they had
developed to solve important, real-world problems had consequences they could not explain. For example,
one such, the Banach{Tarski Paradox, says you can, in principle, take a sphere and cut it up in such a
way that you can reassemble it to form two identical spheres each the same size as the original one.
It became clear, then, that mathematics can lead to realms where the only understanding is through
the mathematics itself. (Because the mathematics is correct, the Banach{Tarski result had to be accepted
as a fact, even though it de es our imagination.) In order to be con dent that we can rely on discoveries
made by way of mathematics|but not veri able by other means|mathematicians turned the methods
of mathematics inwards, and used them to examine the subject itself.
This introspection led, in the middle of the nineteenth century, to the adoption of a new and di erent
conception of the mathematics, where the primary focus was no longer on performing a calculation or
computing an answer, but formulating and understanding abstract concepts and relationships. This was
a shift in emphasis from doing to understanding. Mathematical objects were no longer thought of as
given primarily by formulas, but rather as carriers of conceptual properties. Proving something was no
longer a matter of transforming terms in accordance with rules, but a process of logical deduction from
concepts.
This revolution|for that is what it amounted to|completely changed the way mathematicians
thought of their subject. Yet, for the rest of the world, the shift may as well have not occurred. The
rst anyone other than professional mathematicians knew that something had changed was when the
new emphasis found its way into the undergraduate curriculum. If you, as a college math student, nd
yourself reeling after your rst encounter with this \new math," you can lay the blame at the feet of
the mathematicians Lejeune Dirichlet, Richard Dedekind, Bernhard Riemann, and all the others who
ushered in the new approach.
As a foretaste of what is to come, I'll give one example of the shift. Prior to the nineteenth century,
mathematicians were used to the fact that a formula such as y = x2 + 3x 5 speci es a function that
produces a new number y from any given number x. Then the revolutionary Dirichlet came along and
said, forget the formula and concentrate on what the function does in terms of input{output behavior. A
function, according to Dirichlet, is any rule that produces new numbers from old. The rule does not have
to be speci ed by an algebraic formula. In fact, there's no reason to restrict your attention to numbers.
A function can be any rule that takes objects of one kind and produces new objects from them.
This de nition legitimizes functions such as the one de ned on real numbers by the rule:
If x is rational, set f(x) = 0; if x is irrational, set f(x) = 1:
Try graphing that monster!
Mathematicians began to study the properties of such abstract functions, speci ed not by some formula
but by their behavior. For example, does the function have the property that when you present it with
di erent starting values it always produces di erent answers? (This property is called injectivity.)
This abstract, conceptual approach was particularly fruitful in the development of the new subject
called real analysis, where mathematicians studied the properties of continuity and di erentiability of
functions as abstract concepts in their own right. French and German mathematicians developed the
\epsilon-delta de nitions" of continuity and di erentiability, that to this day cost each new generation of
post-calculus mathematics students so much e ort to master.
Again, in the 1850s, Riemann de ned a complex function by its property of di erentiability, rather
than a formula, which he regarded as secondary.
The residue classes de ned by the famous German mathematician Karl Friedrich Gauss (1777{1855),
which you are likely to meet in an algebra course, were a forerunner of the approach|now standard|
whereby a mathematical structure is de ned as a set endowed with certain operations, whose behaviors
are speci ed by axioms.
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) BACKGROUND READING 5
Taking his lead from Gauss, Dedekind examined the new concepts of ring, eld, and ideal|each of
which was de ned as a collection of objects endowed with certain operations. (Again, these are concepts
you are likely to encounter soon in your post-calculus mathematics education.)
And there were many more changes.
Like most revolutions, the nineteenth century change had its origins long before the main protagonists
came on the scene. The Greeks had certainly shown an interest in mathematics as a conceptual endeavor,
not just calculation, and in the seventeenth century, calculus co-inventor Gottfried Leibniz thought deeply
about both approaches. But for the most part, until the nineteenth century, mathematics was viewed
primarily as a collection of procedures for solving problems. To today's mathematicians, however, brought
up entirely with the post-revolutionary conception of mathematics, what in the nineteenth century was
a revolution is simply taken to be what mathematics is. The revolution may have been quiet, and to a
large extent forgotten, but it was complete and far reaching. And it sets the scene for this book, the
main aim of which is to provide you with the basic mental tools you will need to enter this new world of
modern mathematics (or at least to learn to think mathematically).
Although the post-nineteenth century conception of mathematics now dominates the eld at the post-
calculus, college level, it has not had much in
uence on high school mathematics|which is why you need
a book like this to help you make the transition. There was one attempt to introduce the new approach
into school classrooms, but it went terribly wrong and soon had to be abandoned. This was the so-called
\New Math" movement of the 1960s. What went wrong was that by the time the revolutionaries' message
had made its way from the mathematics departments of the leading universities into the schools, it was
badly garbled.
To mathematicians before and after the mid 1800s, both calculation and understanding had always
been important. The nineteenth century revolution merely shifted the emphasis regarding which of the
two the subject was really about and which played the derivative or supporting role. Unfortunately,
the message that reached the nation's school teachers in the 1960s was often, \Forget calculation skill,
just concentrate on concepts." This ludicrous and ultimately disastrous strategy led the satirist (and
mathematician) Tom Lehrer to quip, in his song New Math, \It's the method that's important, never
mind if you don't get the right answer." After a few sorry years, \New Math" (which was already over
a hundred years old, note) was largely dropped from the school syllabus.
Such is the nature of educational policy making in free societies, it is unlikely such a change could
ever be made in the foreseeable future, even if it were done properly the second time around. It's also not
clear (at least to me) that such a change would be altogether desirable. There are educational arguments
(which in the absence of hard evidence either way are hotly debated) that say the human mind has to
achieve a certain level of mastery of computation with abstract mathematical entities before it is able to
reason about their properties.
4 Why are you having to learn this stu ?
It should be clear by now that the nineteenth century shift from a computational view of mathematics
to a conceptual one was a change within the professional mathematical community. Their interest, as
professionals, was in the very nature of mathematics. For most scientists, engineers, and others who make
use of mathematical methods in their daily work, things continued much as before, and that remains the
same today. Computation (and getting the right answer) remains just as important as ever, and even
more widely used than at any time in history.
As a result, to anyone outside the mathematical community, the shift looks more like an expansion
of mathematical activity than a change of focus. Instead of just learning procedures to solve problems,
college-level math students today also (i.e., in addition) are expected to master the underlying concepts
and be able to justify the methods they use.
Is it reasonable to require this? Granted that the professional mathematicians|whose job it is to
develop new mathematics and certify its correctness|need such conceptual understanding, why make
it a requirement for those whose goal is to pursue a career in which mathematics is merely a tool?
(Engineering for example.)
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) BACKGROUND READING 6
There are two answers, both of which have a high degree of validity. (spoiler: It only appears that
there are two answers. On deeper analysis, they turn out to be the same.)
First, education is not solely about the acquisition of speci c tools to use in a subsequent career.
As one of the greatest creations of human civilization, mathematics should be taught alongside science,
literature, history, and art in order to pass along the jewels of our culture from one generation to the
next. We humans are far more than the jobs we do and the careers we pursue. Education is a preparation
for life, and only part of that is the mastery of speci c work skills.
That rst answer should surely require no further justi cation. The second answer addresses the
tools-for-work issue.
There is no question that many jobs require mathematical skills. Indeed, in most industries, at
almost any level, the mathematical requirements turn out to be higher than is popularly supposed, as
many people discover when they look for a job and nd their math background lacking.
Over many years, we have grown accustomed to the fact that advancement in an industrial society
requires a workforce that has mathematical skills. But if you look more closely, those skills fall into
two categories. The rst category comprises people who, given a mathematical problem (i.e., a problem
already formulated in mathematical terms), can nd its mathematical solution. The second category
comprises people who can take a new problem, say in manufacturing, identify and describe key features
of the problem mathematically, and use that mathematical description to analyze the problem in a precise
fashion.
In the past, there was a huge demand for employees with type 1 skills, and a small need for type 2
talent. Our mathematics education process largely met both needs. It has always focused primarily on
producing people of the rst variety, but some of them inevitably turned out to be good at the second kind
of activities as well. So all was well. But in today's world, where companies must constantly innovate to
stay in business, the demand is shifting toward type 2 mathematical thinkers|to people who can think
outside the mathematical box, not inside it. Now, suddenly, all is not well.
There will always be a need for people with mastery of a range of mathematical techniques, who
are able to work alone for long periods, deeply focused on a speci c mathematical problem, and our
education system should support their development. But in the twenty- rst century, the greater demand
will be for type 2 ability. Since we don't have a name for such individuals (\mathematically able" or even
\mathematician" popularly imply type 1 mastery), I propose to give them one: innovative mathematical
thinkers.
This new breed of individuals (well, it's not new, I just don't think anyone has shone a spotlight on
them before) will need to have, above all else, a good conceptual (in an operational sense) understanding
of mathematics, its power, its scope, when and how it can be applied, and its limitations. They will also
have to have a solid mastery of some basic mathematical skills, but that skills mastery does not have to
be stellar. A far more important requirement is that they can work well in teams, often cross-disciplinary
teams, they can see things in new ways, they can quickly learn and come up to speed on a new technique
that seems to be required, and they are very good at adapting old methods to new situations.
How do we educate such individuals? We concentrate on the conceptual thinking that lies behind all
the speci c techniques of mathematics. Remember that old adage, \If you give a man a sh you can
keep him alive for a day, but if you teach him how to sh he can keep himself alive inde nitely"? It's the
same for mathematics education for twenty- rst century life. There are so many di erent mathematical
techniques, with new ones being developed all the time, that it is impossible to cover them all in K-16
education. By the time a college frosh graduates and enters the workforce, many of the speci c techniques
learned in those four college-years are likely to be no longer as important, while new ones are all the rage.
The educational focus has to be on learning how to learn.
The increasing complexity in mathematics led mathematicians in the nineteenth century to shift
(broaden, if you prefer) the focus from computational skills to the underlying, foundational, conceptual
thinking ability. Now, 150 years later, the changes in society that were facilitated in part by that more
complex mathematics, have made that focal shift important not just for professional mathematicians but
for everyone who learns math with a view to using it in the world.
So now you know not only why mathematicians in the nineteenth century shifted the focus of math-
ematical research, but also why, from the 1950s onwards, college mathematics students were expected
to master conceptual mathematical thinking as well. In other words, you now know why your college
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) BACKGROUND READING 7
or university wants you to take that transition course, and perhaps work your way through this book.
Hopefully, you also now realize why it can be important to you in living your life, beyond the immediate
need of surviving your college math courses.
NOTE: This course reading is abridged from the course textbook, Introduction to Mathematical Thinking, by me (Keith
Devlin), available from Amazon as a low-cost, print-on-demand book. You don't need to purchase the book to complete the
course, but I know many students like to have a complete textbook. In developing this course, I rst wrote the textbook,
and then used it to construct all the course materials.
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) SUPPLEMENT 1
Elements of Set Theory
This course assumes students have learned basic set theory. This brief document summarizes what is required.
The concept of a set is extremely basic and pervades the whole of present-day mathematical thought.
Any well-de ned collection of objects is a set. For instance we have:
the set of all students in your class
the set of all prime numbers
the set whose only member is you.
All it takes to determine a set is some way of specifying the collection. (Actually, that is not correct. In
the mathematical discipline called abstract set theory, arbitrary collections are allowed, where there is
no de ning property.)
If A is a set, then the objects in the collection A are called either the members of A or the elements
of A. We write
x 2 A
to denote that x is an element of A.
Some sets occur frequently in mathematics, and it is convenient to adopt a standard notation for
them:
N : the set of all natural numbers (i.e., the numbers 1, 2, 3, etc.)
Z : the set of all integers (0 and all positive and negative whole numbers)
Q : the set of all rational numbers (fractions)
R : the set of all real numbers
Thus, for example,
x 2 R
means that x is a real number. And
(x 2 Q) ^ (x > 0)
means that x is a positive rational number.
There are several ways of specifying a set. If it has a small number of elements we can list them. In
this case we denote the set by enclosing the list of the elements in curly brackets; thus, for example,
f1; 2; 3; 4; 5g
denotes the set consisting of the natural numbers 1, 2, 3, 4 and 5.
By use of `dots' we can extend this notation to any nite set; e.g.
f1; 2; 3; : : : ; ng
denotes the set of the rst n natural numbers. Again
f2; 3; 5; 7; 11; 13; 17; : : : ; 53g
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) SUPPLEMENT 2
could (given the right context) be used to denote the set of all primes up to 53.
Certain in nite sets can also be described by the use of dots (only now the dots have no end), e.g.
f2; 4; 6; 8; : : : ; 2n; : : :g
denotes the set of all even natural numbers. Again,
f: : : ;8;6;4;2; 0; 2; 4; 6; 8; : : :g
denotes the set of all even integers.
In general, however, except for nite sets with only a small number of elements, sets are best described
by giving the property which de nes the set. If A(x) is some property, the set of all those x which satisfy
A(x) is denoted by
fx j A(x)g
Or, if we wish to restrict the x to those which are members of a certain set X, we would write
fx 2 X j A(x)g
This is read \the set of all x in X such that A(X)". For example:
N = fx 2 Z j x > 0g
Q = fx 2 R j (9m; n 2 Z)[(m > 0) ^ (mx = n)]g
f
p
2;
p
2g = fx 2 R j x2 = 2g
f1; 2; 3g = fx 2 N j x < 4g Two sets, A;B are equal, written A = B, if they have exactly the same elements. As the above example shows, equality of sets does not mean they have identical de nitions; there are often many di erent ways of describing the same set. The de nition of equality re ects rather the fact that a set is just a collection of objects. If we have to prove that the sets A and B are equal, we usually split the proof into two parts: (a) Show that every member of A is a member of B. (b) Show that every member of B is a member of A. Taken together, (a) and (b) clearly imply A = B. (The proof of both (a) and (b) is usually of the `take an arbitrary element' variety. To prove (a), for instance, we must prove (8x 2 A)(x 2 B); so we take an arbitrary element x of A and show that x must be an element of B.) The set notations introduced have obvious extensions. For instance we can write Q = fm=n j m; n 2 Z; n 6= 0g and so on. It is convenient in mathematics to introduce a set which has no elements: the empty set (or null set). There will only be one such set, of course, since any two such will have exactly the same elements and thus be (by de nition) equal. The empty set is denoted by the Scandinavian letter ; [Note that this is not the Greek letter .] The empty set can be speci ed in many ways; e.g. ; = fx 2 R j x2 < 0g ; = fx 2 N j 1 < x < 2g ; = fx j x 6= xg KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) SUPPLEMENT 3 Notice that ; and f;g are quite di erent sets. ; is the empty set: it has no members. f;g is a set which has one member. Hence ; 6= f;g What is the case here is that ; 2 f;g (The fact that the single element of f;g is the empty set is irrelevant in this connection: f;g does have an element, ; does not.) A set A is called a subset of a set B if every element of A is a member of B. For example, f1; 2g is a subset of f1; 2; 3g. We write A B to mean that A is a subset of B. If we wish to emphasize that A and B are unequal here, we write A B and say that A is a proper subset of B (This usage compares with the ordering relations and < on R.) Clearly, for any sets A;B, we have A = B i (A B) ^ (B A) Exercises 1 1. What well-known set is this: fn 2 N j (n > 1) ^ (8x; y 2 N)[(xy = n) ) (x = 1 _ y = 1)]g
2. Let
P = fx 2 R j sin(x) = 0g ; Q = fn j n 2 Zg
What is the relationship between P and Q ?
3. Let
A = fx 2 R j (x > 0) ^ (x2 = 3)g
Give a simpler de nition of the set A.
4. Prove that for any set A:
; A and A A
5. Prove that if A B and B C, then A C
6. List all subsets of the set f1; 2; 3; 4g.
7. List all subsets of the set f1; 2; 3; f1; 2gg.
8. Let A = fx j P(x)g;B = fx j Q(x)g, where P;Q are formulas such that 8x[P(x) ) Q(x)]. Prove
that A B.
9. Prove that A B if and only if P(A) P(B).
10. Prove (by induction) that a set with exactly n elements has 2n subsets.
11. Let
A = fo; t; f; s; e; ng
Give an alternative de nition of the set A. (Hint: this is connected with N but is not entirely
mathematical.)
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) SUPPLEMENT 4
There are various natural operations we can perform on sets. (They correspond roughly to addition,
multiplication, and negation for integers.)
Given two sets A;B we can form the set of all objects which are members of either one of A and B.
This set is called the union of A and B and is denoted by
A [ B
Formally, this set has the de nition
A [ B = fx j (x 2 A) _ (x 2 B)g
(Note how this is consistent with our decision to use the word `or' to mean inclusive-or.)
The intersection of the sets A;B is the set of all members which A and B have in common. It is
denoted by
A \ B
and has the formal de nition
A \ B = fx j (x 2 A) ^ (x 2 B)g
Two sets A;B are said to be disjoint if they have no elements in common: that is, if A \ B = ;.
The set-theoretic analog of negation requires the concept of a universal set. Often, when we are
dealing with sets, they all consist of objects of the same kind. For example, in number theory we may
focus on sets of natural numbers or sets of rationals; in real analysis we usually focus on sets of reals. A
universal set for a particular discussion is simply the set of all objects of the kind being considered. It is
frequently the domain over which the quanti ers range.
Once we have xed a universal set we can introduce the notion of the complement of the set A.
Relative to the universal set U, the complement of a set A is the set of all elements of U that are not in
A. This set is denoted by A0, and has the formal de nition
A0 = fx 2 U j x 62 Ag
[Notice that we write x 62 A instead of :(x 2 A), for brevity.]
For instance, if the universal set is the set N of natural numbers, and E is the set of even (natural)
numbers, then E0 is the set of odd (natural) numbers.
The following theorem sums up the basic facts about the three set operations just discussed.
Theorem Let A;B;C be subsets of a universal set U.
(1) A [ (B [ C) = (A [ B) [ C
(2) A \ (B \ C) = (A \ B) \ C
((1) and (2) are the associative laws)
(3) A [ B = B [ A
(4) A \ B = B \ A
((3) and (4) are the commutative laws)
(5) A [ (B \ C) = (A [ B) \ (A [ C)
(6) A \ (B [ C) = (A \ B) [ (A \ C)
((5) and (6) are the distributive laws)
(7) (A [ B)0 = A0 \ B0
(8) (A \ B)0 = A0 [ B0
((7) and (8) are called the De Morgan laws)
(9) A [ A0 = U
KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2012) SUPPLEMENT 5
(10) A \ A0 = ;
((9) and (10) are the complementation laws)
(11) (A0)0 = A
(self-inverse law)
Proof: Left as an exercise.
Exercises 2
1. Prove all parts of the above theorem.
2. Find a resource that explains Venn diagrams and use them to illustrate and help you understand
the above theorem.
NOTE: This supplement is abridged from the course textbook, Introduction to Mathematical Thinking, by me (Keith
Devlin), available from Amazon as a low-cost, print-on-demand book. You don't need to purchase the book to complete the
course, but I know many students like to have a complete textbook. In developing this course, I rst wrote the textbook,
and then used it to construct all the course materials.
Videos (inc. lectures)
Course assignments
At the end of each lecture, the associated course assignment will be listed here, as a downloadable PDF file.
The assignments are intended to guide understanding of what has been learned. Worked solutions to problems from the two weekly assignments will be demonstrated (video) or distributed (PDFs) in a tutorial session released at 10:00AM PST the following Wednesday (so in weeks 2 through 6).
Working on these assignment problems forms the heart of the learning process in this course. You are expected to form or join a study group, discuss the assignment problems with others in the group, and share your work with them. I strongly urge you to show your work to others in your group and assess one another’s answers, using the course grading rubric to provide yourself and fellow students a sense of how you are doing. For the final exam, you will be graded by, and grade the work of, other students, randomly (and blindly) assigned. The grading will be based on the rubric, so it will be useful to become familiar with it in advance.
Admittedly, peer grading is not the same as grading by an expert who really knows the material. But it is better than nothing! In fact, done conscientiously, using a well designed rubric, it's a lot better than you might think, particularly when the results are compared with grading by an instructor who has a large number of assignments to grade in a limited amount of time! In some studies, students were observed to learn better when they were asked to actively assess their answers and those of their peers according to the instructor's rubric. In particular, students who self-graded using a rubric outperformed students who were graded by instructors. (Sadler and Good, 2006)
The assignments are intended to guide understanding of what has been learned. Worked solutions to problems from the two weekly assignments will be demonstrated (video) or distributed (PDFs) in a tutorial session released at 10:00AM PST the following Wednesday (so in weeks 2 through 6).
Working on these assignment problems forms the heart of the learning process in this course. You are expected to form or join a study group, discuss the assignment problems with others in the group, and share your work with them. I strongly urge you to show your work to others in your group and assess one another’s answers, using the course grading rubric to provide yourself and fellow students a sense of how you are doing. For the final exam, you will be graded by, and grade the work of, other students, randomly (and blindly) assigned. The grading will be based on the rubric, so it will be useful to become familiar with it in advance.
Admittedly, peer grading is not the same as grading by an expert who really knows the material. But it is better than nothing! In fact, done conscientiously, using a well designed rubric, it's a lot better than you might think, particularly when the results are compared with grading by an instructor who has a large number of assignments to grade in a limited amount of time! In some studies, students were observed to learn better when they were asked to actively assess their answers and those of their peers according to the instructor's rubric. In particular, students who self-graded using a rubric outperformed students who were graded by instructors. (Sadler and Good, 2006)
Created Mon 13 Aug 2012 3:10:44 PM PDT
Last Modified Thu 6 Sep 2012 10:21:11 PM PDT
Last Modified Thu 6 Sep 2012 10:21:11 PM PDT
Peer review process
NOTE: This procedure is experimental, and will be under regular review throughout the course. Based on what we observe and learn, we may make adjustments. Any change will be described on the course website.
OVERVIEW
Peer grading is the process of having students grade the assigned work (in our case, the final exam) of other students according to a grading rubric that has been pre-defined by the instructor.
In classes where enrollment numbers are in the thousands, and where student assignments cannot be graded by a computer, peer-grading is the most efficient way of helping students receive scores and feedback.
Though at first it can seem tedious, peer grading is known to have positive effects on learning. Students have been shown to learn better when they are asked to actively compare their answers and that of their peers to the instructor's rubric. In one study on peer grading, students who self-graded outperformed students who were graded by instructors. (Sadler and Good, 2006)
It works like this. As a student you submit your work (in our case, the final exam). Until the grading deadline, you have unlimited chances to re-submit the work, with no penalties for re-submission. When the submission deadline is up, you (and all the other students) undergo a training process in which you go through a small number of grading exercises. Some of these exercises may be anonymous submissions from classmates who have not opted out of the distribution for the grading exercise. Once you have passed the training process, you then peer-grade several submissions from other students in the class, according to the rubric provided. You then self-grade your own assignment, according to the same rubric. Once you have completed peer- and self-grading, you can see your results for the work. You will receive your final grade on the work, as well as the breakdown of each question.
The main goal in having you undergo the training process is to help you become a proficient grader who understand the requirements of the assignment, and can grade a peer's submission within an acceptable range of the instructor's grade on the same assignment.
PEER REVIEW FAQ
Here is a more detailed overview of the various phases of the peer grading exercise. Each phase is compulsory, and requires completion of the previous phase before moving on to the next.
Phase 1 (Complete assignment): In this phase, you take the exam. You will be given a rubric together with instructions. Pay careful attention to the rubric, as later on you will be grading your own exam, as well as your peers, using these metrics. (Even after you have hit submit, you are allowed to re-submit anytime before the exam deadline, with no penalty to your grade. Only your final (re-)submission will be counted when the grading deadline is up.)
Phase 2 (Training): During this phase, you will be given sample solutions to five problems, possibly including actual submissions from other students, to practice grading with.
Passing a grading exercise: In order to pass the grading exercise, you need to grade approximately 80% of the questions within 80% of the instructor's grade. Once you complete a grading exercise, you will be told whether you passed or failed, and will given feedback on your scores.
Passing training: Training consists of a series of up to three grading exercises. Once you pass a grading exercise, you are deemed to have passed training. If you do not pass training by the third attempt, your own submission can still be peer graded and you will receive a grade for your work.
Phase 3 (Grading): Once you have passed training, you will grade five of your classmate's exams. Once you have finished grading these exams, you will move on to self-evaluation.
Self Evaluation: You will be given your own exam solution to grade, using the same rubrics as before.
Phase 4 (Results): In this phase, your final grade and your grade breakdown for each question will be released to you. The grade you receive for each question will be calculated from a combination of your peer-graded score and your self-graded score, according to the formula described below.
OVERVIEW
Peer grading is the process of having students grade the assigned work (in our case, the final exam) of other students according to a grading rubric that has been pre-defined by the instructor.
In classes where enrollment numbers are in the thousands, and where student assignments cannot be graded by a computer, peer-grading is the most efficient way of helping students receive scores and feedback.
Though at first it can seem tedious, peer grading is known to have positive effects on learning. Students have been shown to learn better when they are asked to actively compare their answers and that of their peers to the instructor's rubric. In one study on peer grading, students who self-graded outperformed students who were graded by instructors. (Sadler and Good, 2006)
It works like this. As a student you submit your work (in our case, the final exam). Until the grading deadline, you have unlimited chances to re-submit the work, with no penalties for re-submission. When the submission deadline is up, you (and all the other students) undergo a training process in which you go through a small number of grading exercises. Some of these exercises may be anonymous submissions from classmates who have not opted out of the distribution for the grading exercise. Once you have passed the training process, you then peer-grade several submissions from other students in the class, according to the rubric provided. You then self-grade your own assignment, according to the same rubric. Once you have completed peer- and self-grading, you can see your results for the work. You will receive your final grade on the work, as well as the breakdown of each question.
The main goal in having you undergo the training process is to help you become a proficient grader who understand the requirements of the assignment, and can grade a peer's submission within an acceptable range of the instructor's grade on the same assignment.
PEER REVIEW FAQ
- Where can I find the grading rubric for this course? The rubric will be provided as part of the peer review interface. But you can see it here.
- Can I submit my assignment in any language? No, we cannot support multiple languages at this time; your submission must be in English or you will receive a grade of zero from your peers.
- Will my assignment be anonymous? Please refer to the honor code and privacy policy page.
- Can I opt out of the grading exercise? Course completion does not require participation in the peer review process. However, for this material, you will learn a lot by reviewing the work of others. In order to be receive a grade for the final exam (and thus be eligible for distinction), you must take part in the peer review process.
- If I miss the grading deadline, can I receive late credit? Sorry, at this stage, for logistical reasons, we are not able to extend the deadlines or give late credit in any part of the peer review process.
- Can I get a re-grade on my assignment? Sorry, at this stage, for logistical reasons, we are not able to allow re-grades.
- What do I do if I see a student's assignment containing obscene, hateful, or otherwise abusive content? Contact the staff and the student will be removed from the course.
Here is a more detailed overview of the various phases of the peer grading exercise. Each phase is compulsory, and requires completion of the previous phase before moving on to the next.
Phase 1 (Complete assignment): In this phase, you take the exam. You will be given a rubric together with instructions. Pay careful attention to the rubric, as later on you will be grading your own exam, as well as your peers, using these metrics. (Even after you have hit submit, you are allowed to re-submit anytime before the exam deadline, with no penalty to your grade. Only your final (re-)submission will be counted when the grading deadline is up.)
Phase 2 (Training): During this phase, you will be given sample solutions to five problems, possibly including actual submissions from other students, to practice grading with.
Passing a grading exercise: In order to pass the grading exercise, you need to grade approximately 80% of the questions within 80% of the instructor's grade. Once you complete a grading exercise, you will be told whether you passed or failed, and will given feedback on your scores.
Passing training: Training consists of a series of up to three grading exercises. Once you pass a grading exercise, you are deemed to have passed training. If you do not pass training by the third attempt, your own submission can still be peer graded and you will receive a grade for your work.
Phase 3 (Grading): Once you have passed training, you will grade five of your classmate's exams. Once you have finished grading these exams, you will move on to self-evaluation.
Self Evaluation: You will be given your own exam solution to grade, using the same rubrics as before.
Phase 4 (Results): In this phase, your final grade and your grade breakdown for each question will be released to you. The grade you receive for each question will be calculated from a combination of your peer-graded score and your self-graded score, according to the formula described below.
- How a peer grade is calculated: We will take the median of all the peer grades. The reason for taking the median instead of the average of all peer grades is to reduce unreliable grades – grades that are overly high or low will have a much smaller influence on the final median grade, than if the average were taken.
- If the peer grade and self-grade are within range (that range being determined by the instructor and TAs): we will take the higher grade.
- If the peer grade and self-grade differ significantly, we will take the peer grade to increase grading accuracy.
- Special case: If an instructor or TA has graded an assessment, this grade may override the peer-grade or self-grade.
Created Mon 13 Aug 2012 3:35:14 PM PDT
Last Modified Fri 7 Sep 2012 8:42:38 AM PDT
Last Modified Fri 7 Sep 2012 8:42:38 AM PDT
Course assignments
At the end of each lecture, the associated course assignment will be listed here, as a downloadable PDF file.
The assignments are intended to guide understanding of what has been learned. Worked solutions to problems from the two weekly assignments will be demonstrated (video) or distributed (PDFs) in a tutorial session released at 10:00AM PST the following Wednesday (so in weeks 2 through 6).
Working on these assignment problems forms the heart of the learning process in this course. You are expected to form or join a study group, discuss the assignment problems with others in the group, and share your work with them. I strongly urge you to show your work to others in your group and assess one another’s answers, using the course grading rubric to provide yourself and fellow students a sense of how you are doing. For the final exam, you will be graded by, and grade the work of, other students, randomly (and blindly) assigned. The grading will be based on the rubric, so it will be useful to become familiar with it in advance.
Admittedly, peer grading is not the same as grading by an expert who really knows the material. But it is better than nothing! In fact, done conscientiously, using a well designed rubric, it's a lot better than you might think, particularly when the results are compared with grading by an instructor who has a large number of assignments to grade in a limited amount of time! In some studies, students were observed to learn better when they were asked to actively assess their answers and those of their peers according to the instructor's rubric. In particular, students who self-graded using a rubric outperformed students who were graded by instructors. (Sadler and Good, 2006)
The assignments are intended to guide understanding of what has been learned. Worked solutions to problems from the two weekly assignments will be demonstrated (video) or distributed (PDFs) in a tutorial session released at 10:00AM PST the following Wednesday (so in weeks 2 through 6).
Working on these assignment problems forms the heart of the learning process in this course. You are expected to form or join a study group, discuss the assignment problems with others in the group, and share your work with them. I strongly urge you to show your work to others in your group and assess one another’s answers, using the course grading rubric to provide yourself and fellow students a sense of how you are doing. For the final exam, you will be graded by, and grade the work of, other students, randomly (and blindly) assigned. The grading will be based on the rubric, so it will be useful to become familiar with it in advance.
Admittedly, peer grading is not the same as grading by an expert who really knows the material. But it is better than nothing! In fact, done conscientiously, using a well designed rubric, it's a lot better than you might think, particularly when the results are compared with grading by an instructor who has a large number of assignments to grade in a limited amount of time! In some studies, students were observed to learn better when they were asked to actively assess their answers and those of their peers according to the instructor's rubric. In particular, students who self-graded using a rubric outperformed students who were graded by instructors. (Sadler and Good, 2006)
Created Mon 13 Aug 2012 3:10:44 PM PDT
Last Modified Thu 6 Sep 2012 10:21:11 PM PDT
Last Modified Thu 6 Sep 2012 10:21:11 PM PDT
No comments:
Post a Comment