Math 23a/E-23a
Last Updated on 2nd September 20
Harvard University | Syllabus, Fall 2015
Course Info
Math 23a is the first half of a moderately rigorous course in linear algebra and multivariable calculus, designed for students who
are serious about mathematics and interested in being able to prove the theorems that they use but who are as much concerned
about the application of mathematics in fields like physics and economics as about “pure mathematics” for its own sake. Trying to
cover both theory and practice makes for a challenging course with a lot of material, but it is appropriate for the audience!
Instructor: Paul Bamberg
(to be addressed as “Paul,” please)
Paul graduated from Harvard in 1963 with a degree in physics and received his doctorate in theoretical physics at Oxford in 1967.
He taught in the Harvard physics department from 1967 to 1995 and joined the math department in 2001. From 1982 to 2000 he
was one of the principals of the speech recognition company Dragon Systems. If you count Extension School and Summer School,
he has probably taught more courses, in mathematics, physics, and computer science, than anyone else in the history of Harvard.
He was the first recipient of the White Prize for excellence in teaching introductory physics.
This term, Paul is also teaching Math 152, “Discrete Mathematics,” and Math 116, “Real Analysis, Convexity, and Optimization.”
Email: bamberg@tiac.net
Office: Science Center 322, (617-49)5-9560
Office Hours:
Tuesday and Thursday, 1:30-2:15 in Science Center 322.
Mondays 2-2:30 (longer if students are still there)
Head Teaching Assistant: Kate Penner
(to be addressed as “Kate,” please)
Kate is the course head for Math E-23a, responsible for making it possible for students from around the nation and the world to
participate as fully as possible in course activities.
Kate’s Harvard undergraduate degree is in government, but her interests have moved to political economy and mathematics. After
taking Math E-23 in the Extension School, she became the head teaching assistant and is starting her sixth year on the course staff.
She has been course head for linear algebra and real analysis courses in the Summer School. She may have set a Harvard record
in Spring 2013 by teaching in four courses (Math M, Math 21b, Math 23b, and Math 117). To date, she has received over a dozen
teaching awards from the Bok Center for Teaching and Learning for her work teaching undergraduate math.
This term, Kate is also teaching Math 1a and Math 21a.
Email: penner@math.harvard.edu
Office: Science Center 424
Office Hours: TBD
Course Assistants
Nicolas Campos ncampos@college.harvard.edu
Ju Hyun Lee juhyunlee@college.harvard.edu
Elaine Reichert reichertelaine@gmail.com
Emily Tiberi emilytiberi@college.harvard.edu
Sebastian Wagner-Carena swagnercarena@college.harvard.edu
Jennifer Hu jenniferhu@college.harvard.edu
Michael Lim hyunjaelim@college.harvard.edu
Ben Sorscher bsorscher@college.harvard.edu
Kenneth Wang kwang02@college.harvard.edu
Extension & Distance Team
Giacomo Barbone giacomo.e.barbone@gmail.com
Stu Mason strtmason@gmail.com
Taylor Yeracaris ctay@yyci.com
Jake Carr jake.carr74@gmail.com
Leslie Nordstrom lesliekn@bu.edu
Prerequisites
This course is designed for the student who received a grade of 5 on the Math BC Advanced Placement examination or an A or
A minus in Math 1b. Probably the most important prerequisite is the attitude that mathematics is fun and exciting. Extension
students should ordinarily have an A in Math E-16, and an additional math course would be a very good idea.
Prerequisites, con’t.
Our assumption is that the typical Math 23a student knows only high-school algebra and single-variable calculus, is currently better
at formula-crunching than at doing proofs (may have no proof experience), and likes to see examples to accompany abstractions. If,
before coming to Harvard, you took courses in both linear algebra and multivariable calculus, Math 25 might be more appropriate.
We do not assume that Math 23 students have any prior experience in either of these areas beyond solving systems of linear
equations in high school algebra.
This year, for the second time, we will devote four weeks to single-variable real analysis. Real analysis is the study of real-valued
functions and their properties, such as continuity, and differentiability, as well as sequences, series, limits, and convergence. This
means that if you are an international student whose curriculum included calculus but not infinite series OR if you had a calculus
course that touched only lightly on topics like series, limits, and continuity, you will be OK.
Mathematics beyond AP calculus is NOT a prerequisite! Anyone who tries to tell you otherwise is misguided. In fact, since we will
be teaching sequences and series from scratch (but rigorously), you can perhaps get away with a weaker background in this area
than is required for Math 21.
Strange as it may seem, Part I of the math placement test that freshmen have taken is the most important. Students who do well in
Math 23 have almost all scored 26 or more out of 30 on this part.
Extension students who register for graduate credit are required to learn and use the scripting language R. This option is also
available to everyone else in the course. You need to be only an experienced software user, not a programmer.
Who takes Math 23?
When students in Math 23b were asked to list the two concentrations they were most seriously considering, the most popular
choices were mathematics, applied math, physics, computer science, chemistry, mathematical economics, life sciences, and
humanities.
Extension students who take this course are often establishing their credentials for a graduate program in a field like mathematical
economics, mathematics, or engineering. Programs in fields like economics like to see a course in real analysis on your transcript.
Successful Math E-23 students have usually taken more than one course beyond single-variable calculus.
Upperclassmen who have made a belated decision to go into a quantitative PhD program will also find this course useful.
Course Meetings
The course ordinarily meets in Science Center A. To avoid overcrowding, the first two lectures have been moved to
Science Center C.
Lectures on Tuesdays and Thursdays
Lectures run from 2:35 to 4:00. They provide complete coverage of the week’s material, occasionally illustrated by examples done
in the R scripting language.
Problem Sessions (Section)
There are two types of weekly problem sessions led by the course staff. The first is REQUIRED; the second, though highly
recommended, is optional. Students are sectioned into one of each type.
• The “early” sections on Thursday and Friday will be devoted to problem solving in small groups. These are a required course
activity and will count toward your grade. Lecture on Thursday is crucial background for section!
• The “late” sections that meet on Monday will focus on the weekly problem sets due on Wednesday mornings, and will also
review the proofs that were done in lecture. Attendance at these sections is optional, but most students find them to be
time well spent.
Sections will begin on September 10-11. Sectioning will occur in class and via email on Tuesday, Sept. 8. We will post and have
copies of the necessary form to submit to Kate!
In order to include your name on a section list, we must obtain your permission (on the sectioning form) to reveal on the Web site
that you are a student taking Math 23a or E-23a. If you wish to keep this information secret, we will include your name in
alphabetical order, but in the form Xxxx Xxxxxx.
Lecture Videos & Simultaneous Enrollment
Videos will be made of all the lectures. Usually the video will be posted on the Web site before the next lecture, and often it will
appear on the same day. The Thursday video will not be posted in time to provide preparation for the early sections that meet on
Thursdays, and we cannot guarantee that it will appear before the Friday sections.
Even though all lectures are captured on video, Harvard rules forbid undergraduates to register for another course that meets at
the same time as Math 23, even one with just a 30-minute overlap!
Lecture Videos & Simultaneous Enrollment, con’t.
With regard to athletic practices that occur at the same time as classes, policy is less well defined.
The Math 23 policy, based on discussions with the Director of Athletics is: It is OK to take Math 23a and practice for your sport
every Tuesday, but you must not miss Thursday lecture for a practice.
Extension students may choose between attending lecture or watching videos. However, students in Math E-23a who will not
regularly attend lecture on Thursday should sign up for a section that meets as late as possible. Then, with occasional exceptions,
they can watch the video of the Thursday lecture to prepare for section.
Exams
Logistics
There will be two quizzes and one final exam. Quizzes run from 6 to 9 PM, but you can arrive any time before 7 PM, since 120
minutes should be enough time for the quiz.
Quiz 1:
Quiz 2:
Final Exam:
Wednesday, October 7 (module 1, weeks 1-4)
Wednesday, November 4 (module 2, weeks 5-8)
Saturday, December 12, 9AM-noon (module 3, weeks 9-12)
Quizzes are held in Northwest B-103.
KEEP THESE TIME SLOTS OPEN. Do not, for example, schedule a lab or section on Wednesday evenings. If you know that you
tend to work slowly, it would also be unwise to schedule another obligation that leaves only part of that time available to you!
If you have an unexpected time conflict for one of the quizzes, contact Kate as soon as you know about it, and special arrangements
can be made. Distance students will take their quizzes near their home but on the same dates.
Students who have exam accommodations, properly documented by a letter from the Accessible Education Office, may need to take
their quizzes in a separate location. Please provide the AEO letters as early in the term as you can, since we may need to reserve
one or more extra rooms.
A note on course switching
The last day to drop and add courses (like Math 23a and Math 21a) is Monday, October 5. This is BEFORE the first quiz. It is
important that you be aware of how you are managing the material and performing in the course. It is not a good idea to leave
switching out of any course (not just Math 23) until the fifth Monday. Decisions of this nature are best dealt with in as timely a
manner as possible, and under the advisement of course staff!! Please contact us if you find yourself in this situation.
Textbooks
Vector Calculus, Linear Algebra, and Differential Forms, Hubbard and Hubbard, fourth edition, Matrix Editions, 2009. Try to get
the second printing, which includes a few significant changes to chapters 4 and 6.
This book is in stock at the Coop, or you can order it for $84 plus $10 for priority shipping from the publisher’s Web site at
http://matrixeditions.com/UnifiedApproach4th.html.
We will cover Chapters 1-3 this term, Chapters 4-6 in Math 23b; so this one textbook will last for the entire year.
Ross, Elementary Analysis: The Theory of Calculus, 2nd Edition, 2013.
This will be the primary text for the module on single-variable real analysis. It is available electronically through the Harvard library
system (use HOLLIS and search for the author and title). If you like to own bound volumes, used copies can be found on amazon.com
for as little as $25, but be sure to get the correct edition!
Lawvere, Conceptual mathematics: a first introduction to categories, 2nd Edition, 2009.
We will only be using the first chapter, and the book is available for free download through the Harvard library system.
Proofs & the Proof Logger
Presenting Proofs
Learning proofs can be fun, and we have put a lot of work into designing an enjoyable way to learn high level and challenging mathematics! Each week’s course materials includes two proofs. Often these proofs appear in the textbook and will also be covered in
lecture.
You, as students, will earn points towards your grade by presenting these proofs to teaching staff and to each other without the aid
of your course notes. Here is how the system works:
When we first learn a proof in class, only members of the teaching staff are “qualified listeners.” Anyone who presents a satisfactory
proof to a qualified listener also becomes qualified and may listen to proofs by other students. This process of presenting proofs to
qualified listeners occurs separately for every proof.
You are expected to present each proof before the date of the quiz on which it might appear; so each proof has a deadline date.
Each proof is worth 1 point. Here is the grading system:
• Presenting a proof to Paul, Kate, one of the course assistants, or a fellow student who has become a qualified listener: 0.95
points before the deadline, 0.8 points after the deadline. You may only present each proof once.
• Listening to a fellow student’s proof: 0.1 point. Only one student can receive credit for listening to a proof.
Students who do the proofs early and listen to lots of other students’ proofs can get more than 100%, but there is a cap of 30 points
total.You can almost reach this cap by doing each proof before the deadline and listening twice to each proof.
Either you do a proof right and get full credit, or you give up and try again later. There is no partial credit. It is OK for the listener to
give a couple of small hints.
You may consult the official list of proofs that has the statement of each theorem to be proved, but you may not use notes. That
will also be the case when proofs appear on quizzes and on the final exam.
The Proof Logger
It is your responsibility to use the proof logging software on the course Web site to keep a record of proofs that you present or
listen to. You can also use the proof logging software to announce proof parties and to find listeners for your proofs. We will go
over how to sign up for an account later in the semester.
Homework
Homework (typically 8 problems) will be assigned weekly. The assignment will be included in the same online document as the
lecture notes and section problems.
Assignments are due on WEDNESDAYS by 10:00 AM. There will be a locked box on the second floor, near Room 209, with your
“late” section instructor’s name. At 10 AM Kate will place a sheet of colored paper in each box, and anything above that paper will
be late! Please include your name, the assignment number, and your CA’s name on your assignment.
Each week’s assignment will include a couple of optional problems whose solutions require R scripts. These scripts should be uploaded electronically to the dropbox on the Web site for that week. Please include your name as a comment in the script and also
in the file name.
The course assistant who leads your “late” section will return your corrected homework to you at the section after the due date. If
you are not receiving graded homework on schedule, send an email to penner@math.harvard.edu and the problem will be dealt
with.
Homework that is handed in after 10AM on the Wednesday when it is due will not be graded. If it arrives before the end of Reading
Period and looks fairly complete, you will get a grade of 50% for it.
Collaboration & Academic Integrity Policy
You are encouraged to discuss the course with other students and with the course staff, but you must always write your homework
solutions out yourself in your own words. You must write the names of those you’ve collaborated with at the top of your assignment.
If you collaborate with classmates to solve problems that call for R scripts, create your own file after your study group has figured
out how to do it.
Proofs that you submit to the course Web site must be done without consulting files that other students have posted!
If you have the opportunity to see a complete solution to an assigned problem, please refrain from doing so. If you cannot resist the
temptation, you must cite the source, even if all that you do is check that your own answer is correct.
You are forbidden to upload solutions to homework problems, whether your own or ones that are posted on the course Web site,
to any publicly available location on the Internet.
Anything that you learn from lecture, from the textbook, or from working homework problems can be regarded as “common knowledge” for purposes of this course, and the source need not be cited. Anything learned in prerequisite courses falls into the same
category. Do not assume that other courses use such an expansive definition of “common knowledge”!
Honor Code
If you are a freshman, you will already be familiar with the new Harvard College Honor code, which states:
“Members of the Harvard College community commit themselves to producing academic work of integrity – that is, work that
adheres to the scholarly and intellectual standards of accurate attribution of sources, appropriate collection and use of data, and
transparent acknowledgement of the contribution of others to their ideas, discoveries, interpretations, and conclusions. Cheating
on exams or problem sets, plagiarizing or misrepresenting the ideas or language of someone else as one’s own, falsifying data, or
any other instance of academic dishonesty violates the standards of our community, as well as the standards of the wider world of
learning and affairs.”
On the final exam, everyone will be asked to affirm their awareness of the Honor Code. Here is the current statement of the policy:
“At seated final exams, all students will be asked to read and sign a statement affirming their awareness of the Honor Code. Faculty
will be provided with attendance slips that include the affirmation in their exam packets (”I affirm my awareness of the standards
of the Harvard College Honor Code.”). Faculty will be asked to distribute these slips with the exams and will ask students to sign
the affirmation before completing the exam. Students will turn in the slips with their exam. Faculty may also have the option of
including the affirmation on the printed exam and asking students to sign the affirmation directly there. Faculty may also ask
students to hand write the affirmation and sign it directly on their blue book.
Extension students and students cross-registered from the Kennedy School, the School of Public Health, and other graduate
schools are “members of the Harvard College community” while participating in Math 23a/E-23a and are expected to conform
to the standards of the Honor Code.
For purposes of quizzes and exams, this mostly means “no cheating,” even if you are taking a quiz as a distance learner. The issues
of citing sources and acknowledging the contributions of others are important for homework and for graduate term projects in R.
However, on exams this issue should not arise, since everything presented in lectures, videos, R scripts, the textbook, notes, etc. is
deemed “common knowledge,” and common knowledge does not have to be cited.
Getting Help: Tutoring
Several excellent students from previous years, qualified to be course assistants but too busy, are registered with the Bureau of
Study Counsel as tutors. If you find yourself getting into difficulties, immediately contact the BSC and get teamed up with one of
them.
You will have to contact the BSC directly to arrange for a tutor, since privacy law forbids anyone on the Math 23 staff to know who
is receiving tutoring. A website with more information can be found at www.bsc.harvard.edu.
BSC tutors should provide general guidance: they should not be “collaborators” in doing homework. Therefore you should never
get into a situation where the requirement of listing collaborators conflicts with privacy law.
Software
LaTeX
Pronounced ”LAY-tek,” this is the technology that is used to create all the course handouts. Once you learn how to use it, you can
create professional-looking mathematics on your own computer. The editor that is built into the Canvas course Web site is based
on LaTeX.
One of the course requirements is to upload four proofs to the course Web site in a medium of your choice. One option is to use
LaTeX. Alternatively, you can use the Canvas file editor (LaTeX based), or you can make a YouTube video.
I learned LaTeX without a book or manual by just taking someone else’s files, ripping out all the content, and inserting my own, and
so can you. You will need to download freeware MiKTeX version 2.9 (see http://www.miktex.org), which includes an integrated
editor named TeXworks.
From http://tug.org/mactex/ you can download a similar package for the Mac OS X.
When in TeXworks, use the Typeset/pdfLaTeX menu item button to create a .pdf file. To learn how to create fractions, sums,
vectors, etc., just find an example in the lecture outlines and copy what I did. All the LaTeX source for lecture outlines, assignments,
and practice quizzes is on the Web site, so you can find working models for anything that you need to do.
The course documents contain examples of diagrams created using TikZ, LaTeX’s built-in graphics editor. It is also easy to include
.jpg or .png files in LaTeX. Students have found numerous other solutions to the problem of creating graphics, so just experiment.
Google search offers up a treasure trove of LaTeX advice.
LaTeX, con’t.
If you create a .pdf file for your homework, please ask your CA if they prefer a digital or hard copy. By default, undergraduates and
“local” Extension students may submit the assignment electronically only if you are out of town on the due date. Individual section
instructors may adopt a more liberal policy about allowing electronic submission. If submitting electronically, you may NOT submit
.tex files! Only .pdfs are acceptable.
R and R Studio
This is required only for Extension students who register for graduate credit, but it is an option for everyone. Consider learning R
if you
• are interested in computer science and want practice in using software to do things that are more mathematical than can be
dealt with in CS 50 or 51.
• are thinking of taking a statistics course, which is likely to use R.
• are hoping to get an interesting summer job or summer internship that uses mathematics or deals with lots of data.
• want to be able to work with large data files in research projects in any field (life sciences, economics and finance, government,
etc.)
R is free, open-source software. Instructions for download and installation are on the Web site. If interested, you will have the
chance to use R at the first section on Thursday, September 10 or Friday, September 11; so install it right away, preferably on a
laptop computer that you can bring to section.
On the course website are a set of R scripts, with accompanying YouTube videos, that explain how to do almost every topic in the
course by using R. These scripts are optional for undergraduates, but they will enhance your understanding both of mathematics
and of R.
Use of R
You can earn “R bonus points” in three ways:
• By being a member of a group that uploads solutions to section problems that require creation of R scripts. These will be
available most, but not all, weeks. (about 10 points)
• By submitting R scripts that solve the optional R homework problems (again available most, but not all, weeks). (about 20
points)
• By doing a term project in R. (about 20 points)
To do the “graduate credit” grade calculation, we will add in your R bonus points to the numerator of your score. To the denominator, we will add in 95% of your bonus points or 50% of the possible bonus points, whichever is greater. Earning a lot of R points
is essential if you are registered for graduate credit. Otherwise,earning more than half the bonus points is certain to raise your
percentage score a bit, and it can make a big difference if you have a bad day on a quiz or on the final exam.
Grades
Your course grade will be determined as follows:
• problem sets, 50 points. Your worst score will be converted to a perfect score.
• presenting and listening to proofs, 26 points.
• uploading proofs to the Web site, 4 points.
• participation in the “early” sections, based on attendance, preparation, contributions to problem solving, and posting solutions
to the Web site, 10 points.
• two quizzes, 40 points each.
• final exam, slightly more than 60 points.
• OPTIONAL: R bonus points, about 50 points in numerator, 25-45 points in denominator.
For graduate students, only a “graduate” percentage score, using the R bonus points, will be calculated. For everyone else, we will
also calculate an “undergraduate” percentage score, ignoring the R bonus points, and we will use the higher of the two percentage
scores.
Grades, con’t.
The grading scheme is as follows:
Points Grade
94.0%
A
88.0%
A80.0%
B+
75.0%
B
69.0%
B63.0%
C+
57.0%
C
51.0%
CThere is no “curve” in this course! You cannot do worse because your classmates do better.
Week-by-week Schedule
Month
Fortnight 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Week 9
Week 10
Week 11
Fortnight 12
Half-week 13
Date
September 3-11
September15-18
September 22-25
Sept. 29 - Oct. 2
October 6-9
October 7
October 13-16
October 20-23
October 27-30
November 3-6
November 4
November 10-13
November 17-20
Nov. 24-Dec. 3
November 26
December 8
December 12
Topic
Fields, vectors and matrices
Dot and cross products; Euclidean geometry of Rn
Row reduction, independence, basis
Eigenvectors and eigenvalues
Number systems and sequences
QUIZ 1 on weeks 1-4
Series, convergence tests, power series
Limits and continuity of functions
Derivatives, inverse functions, Taylor series
Topology, sequences in Rn , linear differential equations
QUIZ 2 on weeks 5-8
Limits and continuity in Rn ; partial and directional derivatives
Differentiability, Newton’s method, inverse functions
Manifolds, critical points, Lagrange multipliers
Thansksgiving
Calculus on parametrized curves; div, grad, and curl
FINAL EXAM on weeks 9-12
This schedule covers all the math that is needed for Physics 15a, 16, and 15b with the sole exception of surface integrals, which will
be done in the spring. The real analysis in Math 23a alone will be sufficient for most PhD programs in economics, though the most
prestigious programs will want to see Math 23b also. All the mathematics that is used in Economics 1011a will be covered by the
end of the term. The coverage of proofs is complete enough to permit prospective Computer Science concentrators to skip CS 20.
Abstract vector spaces and multiple integration, topics of great importance to prospective math concentrators, have all been moved
to Math 23b.
Miscellany
Switching Courses - College students only
While transfers among Math 21a, 23a, 25a, and 55a are routine, it is important to note that Math 21a focuses on multivariable
calculus, while Math 23a and 25a focus on linear algebra. Math 21b focuses on linear algebra, while Math 23b and 25b focus on
multivariable calculus. Math 21a and b are given every semester, while Math 23a and 25a are fall only with 23b and 25b given spring
only. Ordinarily there is a small fee if you drop a course after the third Monday of the term, but this is waived in the case of math
courses. However, the fifth Monday, October 5, is a firm deadline after which you cannot change courses! If you are considering
courses, please contact Paul or Kate to be advised correctly!
Special material for Physics 15b and Physics 153
Math 23b does an excellent treatment of “vector calculus” (div, grad, and curl) and its relation to differential form fields and the
exterior derivative. Alas, this material is needed in Physics 15b and Physics 153 before we can reach it in Math 23.
Week 13 covers these topics in a manner that relies only on Math 23a, never mentioning multiple integrals. This will be covered
in a special lecture during reading period, and there will be a optional ungraded problem set. If you choose to do this topic, which
physics students last year said was extremely useful, there will be one question about it on the final exam, which you can use to
replace your lowest score on one of the other questions.
If you are not taking Physics 15b or Physics 153, just wait to see this material in Math 23b.
YouTube videos
These were made as part of a rather unsuccessful pedagogical experiment last year. They are quite good, but you will need some
extra time to watch them.
• The Lecture Preview Videos were made by Kate. They cover the so-called Executive Summaries in the weekly course materials, which go over all the course materials, but without proofs or detailed examples.
If you watch these videos (it takes about an hour per week) you will be very well prepared for lecture, and even the most
difficult material will make sense on a first hearing.
Last year’s experiment was unsuccessful because we assumed in lecture that everyone had watched these videos, when in
fact only half the class did so. Those who did not watch them complained, correctly, that the lectures skipped over basic
material in getting to proofs and examples. This year’s lectures will be self-contained, so the preview videos are not required
viewing.
• The R script videos were made by Paul. They provide a line-by-line explanation of the R scripts that accompany each week’s
materials.
Last year’s experiment was unsuccessful because going over these scripts in class was not a good use of lecture time. If you
are doing the “graduate” option, these scripts are pretty much required viewing, although the scripts are so thoroughly commented that just working through them on your own is perhaps a viable alternative.
If you are doing just the “undergraduate” option, you can ignore the R scripts completely.
Math 23a/E-23a
WHAT YOU NEED TO KNOW DURING SHOPPING WEEK
• The only prerequisite is a 5 on the BC Calculus exam or equivalent preparation, and an enthusiasm and interest in learning a
about abstract mathematics.
• You do NOT need experience writing proofs in advance of this course.
• Lecture takes place TTh @ 2:30. The first few pages of the lecture notes will be provided at the first class. The rest are online.
It is your responsibility to print and bring them with you!
• Each student is sectioned into TWO sections. One (the early, problem-solving one) is REQUIRED. The other (focused on review & important topics) is OPTIONAL, though this will be your homework grader, regardless of whether you attend section.
• Sectioning will happen IN CLASS on TUESDAY, Sept. 8 and via email, for those not in attendance (a sectioning document will
be posted to the course website).
• Sections begin THURSDAY, Sept. 10 for early sections and MONDAY, Sept. 14 for late sections.
• The first problem set is due WEDNESDAY, Sept. 16.
• Introductions to the proof logger and LaTeX will be organized later in the semester.
Math E-23a
HOW-TO for DISTANCE STUDENTS
Lecture
Lecture videos will be recorded and posted to the course website within 24 hours of class.
Dedicated Course Staff
Kate will head the course staff and administration for distance learners.
Distance CA: Giacomo Barbone
Giacomo Barbone graduated from Harvard College in 2014. A Math 23 alum, he spent several semesters as a Math 23 course
assistant and is excited to be rejoining the course staff remotely from France. He is currently working towards a PhD in Physics.
Distance CA: Jake Carr
A recent graduate from Belmont High School, Jake excelled in Math 23 last year. He recently course assisted with Kate in her
summer course on vectors & linear algebra in the Extension School’s Masters in Math for Teaching program. He is currently a
freshman, majoring in astrophysics at the University of Illinois at Urbana-Champaign.
Director of Graduate Studies: Stuart Mason
Stuart will be helping students with the R portions of the course. He works as a senior quantitative analyst at a startup working to
streamline and modernize the way the job market interacts with the internet. He uses R in a practical, applied way every day, and is
excited to see it used to illustrate abstract concepts from higher math. Stuart graduated from Harvard in 2014. He concentrated in
engineering sciences and also earned a secondary in applied mathematics. During undergraduate, he worked for four semesters as
a math 1B CA, and was very excited when Paul gave him the opportunity to return to the math department and resume broadening
his mathematical horizons by helping students do the same.
Section
Early Section (attendance mandatory)
Early section, an opportunity to work with other students in small groups, actively solving problems is a MANDATORY class activity.
This will take place through Blackboard Collaborate, and class links for access will be posted to our course website. Students will
be evaluated on their attendance, participation, and their solutions posted to the course website.
Late Section (attendance optional)
Late section offers a ”recitation” style section, a time where a course assistant will guide their class through the important topics of
the week, focus on items of particular difficulty, and any lingering homework questions. Though attendance is optional, this section
assignment will assign the student to the course assistant who will grade their homework. Students taking this course for graduate
credit will find this section particularly helpful in mastering R ”bonus points.” This will take place through Blackboard Collaborate.
Office Hours
Weekly office hours will be held online with the course staff. Office hours will begin the weekend of Saturday, Sept. 12, with a
precise schedule TBD. Office hours may take place via Blackboard Collaborate or Google Hangout, depending on functionality!
Proofs and Proof Logger
Students will create an account with our course prooflogger (to be unveiled later in the semester), to keep track of their proofs
presented to CAs and classmates. Students may use Google Hangout or Skype to present proofs to classmates.
Quizzes
Quizzes will be administered on the same dates as listed in the in-person syllabus. A proctor is required. Both the proctor and
student will sign a statement, confirming that the exam was administered in keeping with the standards of the College’s honor
code.
Final Exam
The final exam is tentatively scheduled for Saturday, Dec. 12 in the morning. Exams will be arranged through the Distance Education
Registrar and Exams Office.