Japheth Wood

Author: japheth

  • Driving with Cats

    I’ve surprised myself this week with many skills that I didn’t know that I (still) had. Students have left campus (some are weathering it out there), and I’m social distancing at home with my family. I’ve quickly set up home offices, even taped ethernet cable between floors to strengthen our bandwidth, and offered advice on email like the following.

    Hi ——,
    I saw that you’re leaving campus. Good luck! Georgia is a long haul, but the drive could be enjoyable.
    I’ve driven with cats, and here are my tips:
    Keep the cat in a travel carrier, don’t let it roam around the car while you’re driving!
    Line the carrier with a bath towel, in case it has to pee. If that happens, stop as soon as you can and clean things up, because cats like their space clean, and will start kicking around the towel.
    Stop so that the cat can have a pee break now and then. 
    Use a leash whenever the cat is out of the car! Cats startle, run, and hide. It will be difficult to get them back into the car if that happens.
    Start early, and don’t drive if you feel drowsy. Pull over and take a cat nap! Schedule in walk breaks.
    When I drive long distances, the first 100 miles is usually the hardest, and then I get acclimated to the long hauls. 
    It’s better to stop and stay in a motel overnight than to keep driving when it’s dark and you’re drowsy. You’ll have a short easy distance to complete in the morning when you’re refreshed. 
    Wash your hands and don’t touch your face, of course! But driving is a form of social isolation, so just keep hygienic when you stop.
    Good luck! Please get back in touch when things are less crazy.
    Japheth

  • 1988 IMO Problem 6

    I found about this legendary problem from a Numberphile video and was intrigued. Go watch the video, even if you have seen the problem before. It’s fun!

    Here’s the problem which is can be found on The Official IMO page.

    1988 IMO #6. Let $a$ and $b$ be positive integers such that $ab + 1$ divides $a^2 + b^2$.
    Show that $\dfrac{a^{2} + b^{2}}{ab + 1}$ is the square of an integer.

    I thought about the problem for some time and initially thought it was about Gaussian primes. But I got nowhere, so I wrote a simple Python program to collect some data. Denoting the ratio $\dfrac{a^{2} + b^{2}}{ab + 1}$ by $q$, here’s what I found (notice that I considered non-negative integers, rather than just strictly positive integers).:

    $$\begin{array}{c||cccccccccc}
    a & 1 & 1 & 2 & 3 & 8 & 27 & 30 & 112 & 240 & 418 \\ \hline
    b & 0 & 1 & 0 & 0 & 2 & 3 & 8 & 30 & 27 & 112 \\ \hline
    q & 1 & 1 & 4 & 9 & 4 & 9 & 4 & 4 & 9 & 4 \\
    \end{array}$$

    More thoughts…
    What struck me about this data were the solutions for $q = 4$. They are adjacent terms in the sequence
    $$0, 2, 8, 30, 112, 418, \dots$$
    After some thought, I realized that this sequence satisfies the recurrence $T_n = 4 T_{n-1} – T_{n-2}$.

    And the solutions for $q = 9$ are adjacent terms in the sequence $$0, 3, 27, 240, \dots$$
    This sequence satisfies the recurrence $T_n = 9 T_{n-1} – T_{n-2}$.

    Could it be, that for a fixed value of $q$, the integer solutions are adjacent terms of a 2nd order linear recurrence $T_n = q T_{n-1} – T_{n-2}$?!?

    Given one solution $(a,b)$ with $a > b$, the recurrence yields the next solution, which is $(q a – b, a)$.

    But the recurrence also determines the previous terms of the sequence: $T_{n-2} = q T_{n-1} – T_n$.

    So the solution before $(a, b)$ would be $(b, q b – a)$. And if we keep generating smaller solutions, the process would have to bottom out somewhere, namely at $(T_1, T_0) = (r, 0)$ for some $r$.

    Computing the ratio $q$ for this smallest solution yields that $q = r^2$, a perfect square, just as claimed in the problem!

    With that intuition, here is what I proved:

    Suppose that $a > b > 0$ are integers, and $\dfrac{a^2+b^2}{ab+1} = q$ is an integer. Define $a’ = b$ and $b’ = qb – a$. Then

    1. The ratio $\dfrac{a’^2+b’^2}{a’b’+1}$ is also equal to $q$.

    2. $q \geq 1$.

    3. $a’ > b’$.

    4. $b’ \geq 0$.

    Thus, given a solution $(a,b)$, we can derive a smaller solution $(a’,b’)$. So the motivation described above is valid; any solution can be stepped down to a minimal solution $(r,0)$ whose $q$-value is the same as for $(a,b)$, and is $q = r^2$.

    Proofs:

    1. The ratio $\dfrac{a’^2+b’^2}{a’b’+1}$ is also equal to $q$.

    Proof: Start by substitution and then use that $(a^2 + b^2) = q(ab+1)$.
    $$\dfrac{a’^2+b’^2}{a’b’+1} = \dfrac{b^2+(qb – a)^2}{b(qb – a)+1} = \dfrac{q^2b^2 – 2qab + (a^2 + b^2)}{qb^2 – ab + 1} = \dfrac{q^2b^2 – 2qab + q(ab+1)}{qb^2 – ab + 1} = \dfrac{q^2b^2 – qab + q}{qb^2 – ab + 1} = q$$

    2. $q \geq 1$. Proof: This is because $q$ is a positive integer!

    3. $a’ > b’$. Proof: Start with $q = \dfrac{a^2+b^2}{ab+1}$ and decrease the denominator to get
    $q < \dfrac{a^2+b^2}{ab} = \dfrac{a}{b} + \dfrac{b}{a} < \dfrac{a}{b} + 1$. Then note that $q < \dfrac{a}{b} + 1$ is equivalent to $b > bq – a$ which is just $a’ > b’$.

    4. $b’ \geq 0$.
    Proof: In part 1 we saw that $\dfrac{b^2+(qb – a)^2}{b(qb – a)+1} = q$. We know that the numerator and the right hand side are positive. So the denominator must also be positive, which means that $b’ = qb-a \geq 0$.

  • BSME Mathematical Games class

    My next visit was to the BSME Mathematical Games, taught by Anna Kis, an experienced math classroom teacher who is currently earning an advanced teaching degree.

    Anna’s class was the most delightful of my visit, full of playful approaches to teaching advanced mathematical concepts to 11 or 12 year olds. After this class, I was very curious what Anna taught for the rest of the semester, since this class session was just packed with interesting ideas and approaches for the classroom.

    Anna’s lesson broke roughly into four parts: common multiples, congruence classes (mod 4), modular arithmetic, and then extensions. She made clever use of sound, color, patterns and drawings. She also created an interactive and engaging mood, and made expert use of questions. I also enjoyed having her jump outside of the lesson at times to highlight specific ideas from the practioner’s perspective.

    Multiples

    Assignment: make color overlays of the 100 chart using different colors for multiples of 3, 4, 5, and 6. Observe patterns when two overlays are superimposed. In which numbers do two given overlays coincide?

    Activity: three people repeat, “North, South, East, West” in turn. Then each person only speaks out loud when they say “North”.

    Activity: volunteers play musical instruments on every 2nd beat, 3rd beat, and 5th beat. Variation: play on 2nd, 3rd, and 6th beats.

    Discussion: what is the connection between the color overlays game and the music game?

    Anna took care not to answer questions, thus allowing students to express and explain the mathematics in their own words. They were frequently asked to explain the math as an 11 or 12 year old would, so some had to express their more sophisticated mathematical knowledge in more common terms, and others had to work to find appropriate language for their observations.

    Congruence Classes (mod 4)

    Anna drew a number line and marked off the integers from 0 to 20.

    Activity: color the multiples of 4 red.

    A natural question arose: “Should we color zero?” Anna handled this question with a question: “What is a multiple of 4?”, and elicited some student definitions. The consensus definition was then applied to zero, and it was colored red.

    Activity: color the other even numbers blue. How can we (as 11 or 12 year olds) define these numbers mathematically?

    The class definition of even numbers that are not multiples of 4 was “4 box + 2”, where “box” could be any integer.

    Question: if we color every 4th number the same color, how many colors would we need? Describe each color number.

    Congruence Arithmetic

    Anna took out some mini post-its in four colors: red, blue, yellow and green, to represent the color numbers

    Explain: yellow + green = red

    Yellow numbers are 1 more than a multiple of 4, greens are 3 more. A generic diagram of yellow + green went up on the board, and group of four dots were circled, leaving spare groups of 1 and 3, which were combined, completing a red number.

    Task: make a question for the others.

    Question: what color is 201? How do you know?

    Extensions

    Activity: Color Pascal’s Triangle with a secret rule. What is it? (It turned out to be coloring modulo 3.)

    Magic trick: three envelopes labeled 1, 2, and 3 contain numbers. A volunteer chooses two envelopes and takes a number from each and adds them. The magician, upon hearing the sum, knows which envelopes were chosen. How is this done?

    Nim Variant: a number line is drawn, and integers from 0 to 40 are marked off. Starting at 0, two players alternate advancing a pawn at least 1 and at most 4 n each turn. Whoever gets to 40 first wins.

    These further variants each opened up to congruence arithmetic ideas from earlier in the lesson, and illustrated their importance and application.

    I could imagine anything from Anna’s class fitting in very well with many US math circles I’ve seen. I would love to see approaches like this making their way into US math classrooms, and maybe BSME alumni will be the ones to do so!

     

  • A BSME school visit

    My second BSME visit was to a partner school. Every week the BSME students have a visit to one of several partner schools around Budapest. The list is very interesting, and includes many English language schools, elite specialized schools, a school for the blind, and many more.

    Visit to Petrik Lajos School
    Marcia Burrell (SUNY Oswego), Japheth Wood (Bard College), Samuel Otten (U. Missouri), and Douglas Mupasiri (U. Northern Iowa) in front of the Petrik Lajos School

    I was part of a visiting team, invited to learn about BSME, and the program arranged a special visit for us to the Petrik Lajos Bilingual Vocational School of Chemistry, Environmental Protection and Information Technology. This school was founded in 1879, so like many Hungarian schools, has an aura of tradition surrounding it. Many of Hungary’s chemists are educated and trained here. We were hosted by Márti Barbarics, who besides teaching math, is working towards a doctorate in math education.

    Márta is piloting a problem-based math curriculum that seeks to actively engage students. This is a curriculum that is based on the Pósa Method, a project that is led by Péter Juhász, another BSME instructor (more on that in a later post). Márta described the project as quite an innovation, as compared to the traditional curriculum.

    This was a curious fact, in light of Hungary’s fame and tradition in using problem solving to nurture the mathematical interest and expertise of some of the world’s most famous mathematicians and scientists! In fact, many in the US math circle movement trace the origins of math circle back to Hungary (http://mathcircle.berkeley.edu/program) while others point to Bulgaria, Romania and Russia.

    Much of the problem solving method that have had a huge impact on US math education in past decades (and problem solving was quite fashionable, for a time) is encapsulated in George Pólya’s 1945 book, How To Solve It (https://en.m.wikipedia.org/wiki/How_to_Solve_It).

    Pólya (http://www-history.mcs.st-andrews.ac.uk/Biographies/Polya.html) ended up at Stanford University. You might be interested in the Stanford Mathematics Problem Book (https://books.google.com.ar/books/about/The_Stanford_Mathematics_Problem_Book.html), or in that the Stanford Math Circle (http://mathcircle.stanford.edu) was founded in 2005.

    As another sidetrack, one of Pólya’s most famous talks is, “Let us teach guessing”, and you might be able to find the video online. The workshops at HCSSiM start off with this exploration, called “The Watermelon Problem”, and I’ve find that it is a wonderful way to get the conversation started. [I also found out this fall from a delightful math seminar talk at Bard College, delivered by Moshe Cohen of Vassar, that this problem appeared in print as, “Cutting the cheese”, by J.L. Woodbridge, as problem E554 in the 1943 American Math Monthly.]

    Back to the visit to the Petrik Lajos School, Márta had prepared a logic lesson for her students, based on some of Raymond Smullyan’s Knights and Knaves problems (in What is the Name of This Book?) She met with us before the lesson and described her students and her concerns that they might be hesitant toward the lesson, as it differently from the typical lecture mode of delivery. She was ready to explain that logic problems really do appear on the standardized exams, which they might find motivating. Also, Petrik Lajos is a dual language school, so the class would be taught in English.

    Márta was partially right! The students did ask those classic questions, like, “will this be on the test?” But soon they were captured by the beautiful problems, and quite engaged. Márta’s students seemed to enjoy discussing each problem, and how to resolve them.

    That’s part of the point. Students get engrossed by these delightful problems, and become open to genuinely learning the mathematics underneath, and really internalize the content. When skillfully used, an experienced math instructor can select and sequence good problems into an effective and enjoyable math course.

    The Hungarian tradition in mathematical problem solving is now in the process of being rolled out to a wider high school audience, complete with educational studies of efficacy.

    I liked what I saw at Petrik Lajos, but was curious to visit more Hungarian math classrooms. In later posts, I’ll write about more experiences I had with BSME classrooms, and two other school visits I was able to arrange independently of BSME.

     

     

     

  • BSME Directed Research on Gender Issues

    I visited the Budapest Semester in Math Education program in October, 2017, and I’ve been eager to share my notes.

    I really like the program, and think that it can be an impactful experience for future math teachers. Especially for those with a very strong math background. For example, junior staff in the HCSSiM program who’d like to explore teaching, would benefit very much.

    The program is still small, but primed to expand quickly, once it becomes better known.

    My first stop in BSME was to Fruzsina Kollányi’s seminar style directed research workshop on gender issues in math education. This workshop is new to BSME, so let’s call it a pilot class.

    The BSME students were researching the gender achievement gap in Hungarian high schools (they call these “gimnázium”), and Fruzsina had arranged for them to have access to testing data from several specialized schools from around the country. The focus of the investigation would be on Hungary’s highest performing students.

    I was able to point them towards similar research from the US:

    The Gender Gap in Secondary School Mathematics at High Achievement Levels: Evidence from the American Mathematics Competitions
    by Glenn Ellison and Ashley Swanson

    https://economics.mit.edu/files/7598

    The BSME students spent most of the class time designing a survey that they planned to sent out to math teachers and math students. The discussion focused on the logistics and design of the survey. I was glad that they were taking on the details head-on, and that everyone was able to contribute ideas to improve the survey process.

    What made Fruzsina’s course real for me was that the students planned to present their findings at a Hungarian Math Education conference later in the semester. They were preparing to submit an abstract, and would soon be working under a deadline to complete their research study and to prepare their presentation.

    Some more information on the instructor: like most BSME instructors, Fruzsina Kollányi is deeply involved in math education in several ways. She’s a high school math teacher at the Budapest Contemporary Dance Academy, an instructor at Skool (http://skool.org.hu/en/, something akin to Girls Who Code), and educates underserved Roma kids with BAGázs (http://www.bagazs.org/en/).

    UPDATE: Fruzsi shared a link to her students’ presentation: https://prezi.com/view/hMv9jqgiqovge3UKgMEu/

    A Prezi! Here’s another great Hungarian invention, an interactive and intuitive graphical way to organize a presentation.

    Also, she plans to investigate assessment and gamification of learning in future seminars.

    By the way, my favorite math websites that make use of gamification are expii.org, artofproblemsolving.com, and brilliant.org. I’ve also become a big fan of Duolingo, which is a language learning site. Do you have a favorite gamification way to learn math?

  • BSME vs BSM

    I arrived in Hungary, and am getting to know the Budapest Semester in Math Education (BSME) program. This program, now in its 3rd year of operation, is for undergraduate (and recent graduates) who are interested in the teaching of mathematics. They currently offer 5 courses that touch on different aspects of math education, and BSME students can also take BSM courses.

    Yes, there is BSME and there is BSM, which is the Budapest Semesters in Math program, now in its 33rd year. They are not the same program, yet they hold classes in the same building, are both coordinated out of St. Olaf College in the US, and are generally nicely coordinated.

    Keleti Pályaudvar
    The Budapest Semester in Math Education program is just a few blocks away from Keleti Pályaudvar, the Eastern Train Station.

    The 5 courses are the following, and I’ll write about them each in future posts.

    Practicum – Visit math classrooms in schools around Budapest, and debrief.

    Discovery Learning: The Pósa Method – Learn about the math instructional method developed by Lajos Pósa.

    Problem Solving in Secondary School Mathematics – Develop problem solving skills, engage in mathematical tasks to foster deep math learning.

    Concept Building through Games and Manipulatives – Just that, developing mathematical thinking through informal and formal hands-on learning.

    Directed Research: Gender Issues in Mathematics – Think of this as a math education REU.

    Tune in later for more details about these courses, and other aspects of BSME!

     

  • The Budapest Semester in Math Education

    I’m traveling to Hungary next week to visit the new Budapest Semester in Math Education (BSME) program. This program is intended for undergraduate students and recent graduates who are broadly interested in teaching mathematics, and promises to share Hungarian insights into mathematics and math education. BSME is already in its 3rd year, and has just announced the addition of a summer program.

    I do feel compelled to admit that I have a fascination with mathematics in other countries. When I taught in Bard College’s Master of Arts in Teaching (MAT) program, my students and I read and discussed The Teaching Gap by Hiebert and Stiegler, Knowing and Teaching Elementary Mathematics by Liping Ma, and we tried our hand at Japanese Lesson Study. Like many math circle leaders around the country, I’m curious about the content and pedagogy of math circles in Eastern European, especially Bulgaria, Hungary, Romania and Russia, where the US math circle movement traces its roots. I’ve also peeked into math textbooks like Singapore Math, and Russian textbooks.

    I do also have a fascination with mathematics in this country. The history of math and math teaching in the US includes the math curricular reform movements like New Math, the Common Core State Standards, and everything in between, and before. It’s important to know about the Inquiry Based Learning (IBL) movement, the summer math “Epsilon Programs”, math contests and student math journals. The debate between “Back to the Basics” and “Conceptual Understanding” is just a small part of the required reading.

    But what can we learn about math teaching from Hungary? I plan to write more about that during my visit to BSME.

  • How to catch a cheater

    I proctored the AIME II contest this week, and caught a cheater. Here are some details and thoughts about the occasion.

    At about 4pm the day before the contest, I started getting emails and phone calls from parents, from tutors, some students, and even my math colleagues at Bard who had been contacted as well, in desperate attempts to contact me. Their children had all planned to take the contest at Kean University in NJ, but for logistic reasons, the contest manager had to cancel, less than 24 hours in advance.

    Up until then, I only expected to proctor one student, since the other two AIME qualifiers from when we hosted the AMC 10B and AMC 12B contests had also qualified on the A schedule a few weeks earlier. But now I imagined a packed room with 20 or 30 AIME students! The AMC response (amcinfo@maa.org) was quick and responsive, and they granted permission to include as many students as I had AIME answer forms for (I counted, and had 20 forms), even though I had registered and paid for only 10 students.

    I passed on the good news to everyone, and quickly drafted out a liability and photo release form for the newcomers, along with detailed instructions on how to drive up to Bard College (more than 2 hours away from New Jersey) the next morning.

    But then, over the next few hours, all of them found closer places, and cancelled, thankful that the worst case scenario – a long drive up to the Hudson Valley – was there if needed, but the drive could be avoided.

    All but one, that is: the cheater. He is a high school freshman at one of New Jersey’s most prestigious public high schools. He was driven up by his math competition tutor, who described himself as a friend of the family. He coaches the cheater, and four other NJ students in competition math. All the rest of the students had also planned to come up to take the AIME II at Bard, but had found more convenient locations overnight.

    They arrived at Bard, and were welcomed in by the local math student (a middle school student) who had qualified, and his mom. When I arrived a little later, I set up the room, and welcomed the mom and the tutor to enjoy Bard College, a wonderfully secluded campus where they could stretch their legs, catch breathtaking views of the Catskill Mountains and the Hudson River, and explore eclectic architecture (including a Frank Gehry building in north campus). Or, they could set up their laptops with Bard’s wifi, and catch up on work. Bard is on spring break this week, so parking is plentiful, and so are study nooks.

    I helped the students bubble in the answer forms with their names and other information (kids these days don’t remember their street addresses), and made sure they understood the AIME contest rules. Then I started the exam by clicking on the countdown timer (Google search for “internet timer”) set to 3 hours.

    Ten minutes later, the cheater pulled a cell phone out of his pocket and put in on his lap. Did he not understand that electronic devices were strictly not allowed on the AIME? I walked over to him and confiscated his phone immediately. I let him continue the exam, because I saw that he hadn’t received any information from the phone.

    In the middle of the exam, he excused himself to go to the bathroom. He returned a few minutes later. Later, with half an hour to go, he excused himself again to go to the bathroom and left the room.

    A minute later, the other student’s mom opened the door to ask me a question. In the hallway, she told me that the NJ student had gone outside. He told her that he needed to get his photo ID, and she wondered if it was okay that her son didn’t have an ID with him. She was worried about that, but I reassured her that since her son had been to the AMC contests at Bard for several years in a row, I recognized him, and no ID was needed.

    Then the other student came back from outside, not from the bathroom. That was strange. Back in the testing room, the NJ student got back to work, and started to open up a folded piece of paper. I stood up and walked over to him, explaining that outside notes are not allowed on the exam, and asked him to hand it over. I unfolded the paper, and found a numbered list of 3 digit numbers. The answer to each AIME problem is a 3 digit number.

    I was shocked at how blatant this student was in his cheating, and how easy he was to catch.

    I decided to let him continue working, mostly to avoid distracting the other student, and also to maintain control over the situation.

    When the time was up, I asked the students to sign the statement on their answer form that all the work on the contest was their own. Both students signed. I collected the answer forms and excused them from the testing room.

    I told the NJ student that I wanted to speak with him and with his tutor, and he went outside to the parking lot. Then I took the opportunity to thank the local student and his mom, and to ask what problems he found interesting. They left soon after that.

    But the NJ student and his tutor didn’t come back. I waited and waited, and then gathered my things and started leaving. Funny – I still had the student’s cell phone and his cheat sheet. Right before I left the building – it seemed about half an hour after the contest ended – the tutor appeared.

    The tutor asked what had happened. We talked, and the tutor explained that he had given the student the answers on a sheet of paper, because he thought the exam was over (it was just after 12pm when I confiscated it).

    He told me that one of his other NJ students had finished the exam early, about 10am, at some NJ location, and texted an image of the exam to him, which he worked on in his car.

    The student showed up soon after that, and I talked with them both. The student admitted to cheating, and said that he wanted the opportunity to take the USAMO. He said that the tutor didn’t know about the cheating, but I don’t believe him. The tutor also claimed not to know about the cheating, even though he admitted to receiving an electronic copy of the contest around 10am, during the exam, working on it, and handing his student a neatly-written list of answers. He didn’t seem at all upset that his student had lied to him, and asked me to give his student a chance and to forgive the cheating.

    It was fascinating to interview a cheater and his accomplice, and to feel in control of the situation. I asked them question after question to collect information to share with the AMC. I was curious if they would show any remorse for what they had done, and so I asked them questions to open opportunities for that. But there was no remorse at all, so I ended the conversation and left.

    The cheater ran after me again to ask for his cell phone back, but I refused to, and told him to email me and I’d ship it back to him.

    The cheating incident is over, but it leaves me with some questions. If you have answers, please leave a comment!

    1. How important is it to protect the cheater’s identity? I shared his name and other information with the AMC, of course. But I could also easily contact his high school through personal contacts. I could publish his name on the web. I could contact the Kean University contest manager, and other university based contact managers in the region to let them know. Should I take any or all of those steps? Or leave it to the AMC to take action.
    2. Is it important to protect the tutor’s identity? I have contact information for another of his students and that student’s father. What action on my part is appropriate? I was offended on many levels that this tutor placed no priority at all on honesty. In fact, he is a key figure in a cheating gang. If he were tutoring my child, I would want to know.
    3. How welcoming should I be of AIME students from outside the Bard Math Circle community? I want to promote and develop a culture of math enrichment in the Mid-Hudson Valley. I want to open up opportunities for students whose schools don’t have a math club, a math team or a math circle. But this cheating incident was very, very disappointing.
    4. What would you do with the cell phone? (I ended up shipping it to his parents by express mail, carefully bubble wrapped and insured. I included a brief letter explaining that the phone had been confiscated from their son during the AIME exam.)

     

     

  • Mathematical Writing

    Forgive me, but I’m often overjoyed to provide feedback on student writing as a math professor. Who knew?! Below are some of my favorite pieces of advice about writing mathematics, and I hope you will find them useful.

    Paul Halmos wrote How to Write Mathematics (original available here), and these are his key points:

    1. Say something. To have something to say is by far the most important ingredient of good exposition.
    2. Speak to someone. Ask yourself who it is that you want to reach.
    3. Organize. Arrange the material so as to minimize the resistance and maximize the insight of the reader.
    4. Use consistent notation. The letters (or symbols) that you use to denote the concepts that you’ll discuss are worthy of thought and careful design.
    5. Write in spirals. Write the first section, write the second section, rewrite the first section, rewrite the second section, write the third section, rewrite the first section, rewrite the second section, rewrite the third section, write the fourth section, and so on.
    6. Watch your language. Good English style implies correct grammar, correct choice of words, correct punctuation, and common sense.
    7. Be honest. Smooth the reader’s way, anticipating difficulties and forestalling them. Aim for clarity, not pedantry; understanding, not fuss.
    8. Remove the irrelevant. Irrelevant assumptions, incorrect emphasis, or even the absence of correct emphasis can wreak havoc.
    9. Use words correctly. Think about and use with care the small words of common sense and intuitive logic, and the specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning.
    10. Resist symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.

    The Math Association of America is now hosting an entire webpage devoted to Mathematical Communication. This looks like it will become a fantastic resource going forward.

    Over at Harvey Mudd College, Francis Su has posted his Guidelines for Good Mathematical Writing. This seems to be part of a wider campus-wide effort in communication. In this post, Rachel Levy explains why Every Math Major Should Take a Public-Speaking Course.

    Francis Su also has an article Some Guidelines for Good Mathematical Writing in the August/September 2015 MAA Focus Newsmagazine (a subscription or MAA membership might be required).

    Annalisa Crannell at Franklin & Marshall College offers A Guide to Writing in Mathematics Classes for undergraduate math students, but the advice is useful for a much wider audience.

    1. Why Should You Have To Write Papers In A Math Class?
    2. How is Mathematical Writing Different?
    3. Following the Checklist
    4. Good Phrases to Use in Math Papers
    5. Helpful Hints for the Computer
    6. Other Sources of Help

    I’ve also been told from several people to consider Serre’s How to Write Mathematics Badly. I found the video, but had a tough time watching it through to the end. Maybe Serre should have taken the public speaking course at Harvey Mudd College?

     

    Dave Richeson from Dickinson College writes about The Nuts and Bolts of Writing Mathematics at his Division by Zero blog. He also has a very nice Checklist for Editing Mathematical Writing, and an example of a poorly written proof for a class exercise.

  • Uniqueness of Factorization

    A few days ago I came across a proof of the Fundamental Theorem of Arithmetic (aka Unique Factorization) in Courant and Robbin’s What is Mathematics that I hadn’t seen it before. I liked it enough to learn it.

    Then another surprise – I saw it again yesterday in Primes and Programming by Peter Giblin, a book that Larry Zimmerman had recommended to a student from the summer high school program.

    The usual proof that I know is based on Euclid, and basically is a proof by Strong Induction. This new proof is by the Principle of Least Element. So the key is to suppose that unique factorization fails, and to reason about the least positive integer \(N\) that has more than one factorization into primes. Even though we’ll show this number doesn’t exist, we can deduce lots of information about it!

    First, some notation. Let’s say that two distinct prime factorizations of \(N\) are

    \[\text{(1)}\qquad N = p_1 p_2 \dots p_r \text{ and } N = q_1 q_2 \dots q_s\]

    Of course, we’ll arrange the primes in non-decreasing order, so that in particular, \(p_1\) and \(q_1\) are the smallest primes in those factorizations.

    The other primes don’t take a big role in what comes next, so let’s write \(P = p_2 \dots p_r\) and \(Q = q_2 \dots q_s\), so that we have

    \[N = p_1 P = q_1 Q.\]

    The first observation is that \(p_1\) and \(q_1\) are different primes, otherwise if they were equal, we could factor them off and then \(N/p_1 = P\) would be a smaller positive integer with two distinct prime factorizations.

    Now that that’s done, let’s assume without loss of generality that \(p_1 < q_1\), and we’ll form a new number:

    \[\text{(2)}\qquad M = (q_1 – p_1) Q\]

    By equation (2), it’s clear that \(M\) is a positive integer that is less than \(N\), and therefore does factor uniquely into primes. Now we rewrite \(M\) as follows:

    \[M = (q_1 – p_1) Q = q_1 Q – p_1 Q = N – p_1 Q = p_1 P – p_1 Q = p_1 (P – Q)\]

    That is, \[\text{(3)}\qquad M = p_1 (P – Q)\]

    We’re almost there. Note that because of equation (3), the prime \(p_1\) is a prime factor of \(M\). Now consider the factorization of \(M\) given in equation (2). Since we were careful to list the primes in non-decreasing order, \(p_1\) can’t be any of the primes in \(Q = q_2 \dots q_s\), and so it must be a factor of \((q_1 – p_1)\). Suppose that \((q_1 – p_1) = p_1 t\). Then solving for \(q_1\), we find that \(q_1 = p_1 (t+1)\). And this is a contradiction, since then \(q_1\) would not be a prime number!