Sunday, April 19, 2020

Corona Post #2: Teaching Online

[#2 because my previous post in March on my online math circle was due to people needing to take their math circles online when the shelter-in-place orders were just starting.]

I've been teaching online for 4 weeks now, two before our spring break and two after. At first I was just trying to learn how to manage teaching on zoom. I bought a whiteboard that's still sitting on two chairs in my living room, and I sit in a tiny chair while I write on it. Not ideal, but I get to see my students, and I feel like I'm still working with them where they are, not just lecturing.

(Some day I'll finally install it on my living room wall. I procrastinate with tasks like that. I'm not sure why it feels intimidating...)













A few weeks ago I made a google slides presentation for my Discrete Math course to explain a way of counting possibilities called Stars & Bars. I had fun doing it. You're welcome to modify it and use it in your teaching.








Just now I made another. This one is for Calculus II, on Taylor & Maclaurin Series (really just a Maclaurin series). I was motivated by knowing that there would be too much writing for my little whiteboard. This presentation has a handout to go with it.








I'm also teaching Calculus I. I haven't made any cool new materials for that course yet. But I will...

Monday, March 23, 2020

Online Math Circle: Pythagorean Triples



The Pythagorean theorem tells us that if a and b are the legs, and c the hypotenuse, of a right triangle, then a2+b2 = c2. Usually that makes at least one side something ugly like square root of 2. But a few combinations make all three sides whole numbers. Those are called Pythagorean triples. Here are a few of them: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 20-12-29.


Are there patterns to this? Let's play, and see what we can figure out! (We will use some algebra.)



Edited to add:

This online math circle happened on Friday, March 27, at 10am PDT (1pm EDT). [This link is to the zoom recording, along with its automatically produced (therefore hilariously bad) audio transcript.]

I promised to write up some of it here.

Way back in 2007, I read Bob and Ellen Kaplan's book, Out of the Labyrinth: Setting Mathematics Free, about the math circles they lead. It was such a discovery for me! I went to their first Summer Math Circle Teacher Training Institute, held at Notre Dame, and fell in love with this community. I kept going back for years, craving a discussion of math among equals, figuring out new ways of seeing. One summer we discussed Pythagorean triples, and that December I tried to rebuild what I had learned. I am blessed with a very bad memory, so what I did in December looked very different from what we had done in the summer.

I was also exploring online, and ended up putting together a book that collected some of the best resources I had found: Playing with Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers.

Our circle was prompted by Rodi Steinig's request for help learning how to use zoom for online math circles. I offered one of my favorite topics, and off we went. Participants came from as far away as Colombia (and farther?).

We proved a few things, and explored a bunch more. I hope some participants went home eager to prove more on their own.

Saturday, January 4, 2020

Multiplication Chart with Pictures

In the story I'm writing, Althea remembers a multiplication chart that was posted in their bathroom. It had cool pictures around the edges for many of the facts.

  • 2x3 was a 6-pack of soda.
  • 2x6 was a carton of eggs.
  • 8x8 was a chessboard.
  • The fives were sometimes collections of nickels, but 5x12 was the 60 minutes on a clock, and 5x6 was time too.

I thought I knew of more iconic sets like these, but I can't think of any more as good as these. I'm hoping for help. Do you have images in your head for any of the multiplication facts?

Maybe threes will be 3-leaf clovers. 6 of them have 18 petals. That doesn't seem nearly as iconic as the ones above, though.

Fours could be legs on dogs. 6 dogs have 24 legs. Twos could be eyes on friends...


What are your favorite images for multiplication facts?

Saturday, September 21, 2019

The History of Imaginary Numbers is a Soap Opera

I first read about this history in Journey Through Genius, by William Dunham, in the chapter titled Cardano and the solution of the cubic. It reads like a soap opera in which the math is done for glory, not for any possible connection to the real world. The people who came up with imaginary numbers thought they were fictitious elements in the process of solving a cubic, and never expected them to have any real meaning. Turns out they do. Imaginary numbers help scientists describe electrical current and probability distributions, among other things.

Years ago, on Living Math Forum, a mom wrote in to ask for help. Her son had asked: “The square root of 1 is 1, so what's the square root of -1 ?” That inspired me to write a math poem, Imaginary Numbers Do the Trick.

Recently I was having a lovely discussion with my editor, Maria Droujkova, and another author, about math storytelling. I realized this topic might possibly make for a good children's story. I'm working on it now. As I think about it, I'm not sure how to find the right age range. The math seems like it requires high school, but the story could interest younger kids, I think.



The History
Here's a short version:
  • Scipione del Ferro solves equations of the form  x3 + mx  = n (called depressed cubics). On his deathbed, in 1506, he passes his method on to his student, Antonio Fior.
  • Niccolo Tartaglia boasts that he can solve cubics of the form x3 + mx2  = n, so in 1535, Fior challenges him with 30 depressed cubics. (These challenges were a common feature of life as a mathematician in 1500's Italy, and provided a way for mathematicians to get more recognition and paying students.) Tartaglia's return problem list to Fior has a variety of problems. Tartaglia does not yet have a solution for the depressed cubic, and sweats it, working feverishly to try to figure it out. At the last moment, he succeeds, and solves all 30 problems. Fior does not do so well, and is humiliated.
  • Gerolamo Cardano comes to Tartaglia, asking him to disclose his method. He begs repeatedly, and Tartaglia, now Cardano's guest in Milan, finally concedes. Cardano takes an oath of secrecy. Tartaglia writes his solution in cipher, as a poem (!). 
  • Cardano takes on a brilliant student, Ludovico Ferrari, with whom he shares the secret. Together, they solve the general cubic, and then Ferrari goes on to solve the quartic. But all their work depends on reducing these to the depressed cubic, which Cardano has sworn not to tell about.
  • Cardano and Ferrari travel to Bologna, and are able to inspect the papers of ... Scipione del Ferro, where they find the solution. Cardano figures that relieves him of his oath and publishes, in his 1545 book, Ars Magna. He gives both del Ferro and Tartaglia credit, but Tartaglia is furious.
  • In the book, Cardano lays out the steps for solving the general cubic. But in doing so, he introduces a mystery. The depressed cubic x3 - 15x  = 4 clearly has solutions x = 4 and x = -2+-√3. And yet the formula found by Ferro, Tartaglia, Cardano, and Ferrari includes a √-121 for this equation. Cardano threw up his hands at the mystery. It was explored but not truly understood 30 years later by Rafael Bombelli. It took another almost two centuries for Euler to finally solve the mysteries of complex numbers.

    Here's a nice write-up I found online, but it suggests different facts than the version in Dunham's book. I will keep reading while I write, so I can hopefully get my facts right.



    My Request
    I'm looking for kids who would like to read my draft versions and tell me what parts they like. If you have kids who understand (at all) the notion of a square root and the idea of what a cubic equation is, would you ask them if they'd like to read my story? (I would, of course, mention them in my book if it gets published.) You, or your kids, can email me at mathanthologyeditor on gmail.



    Just a Beginning
    The Saga of the Imaginary Numbers
    “Mom, I’ve been thinking… If the square root of 1 is 1, what is the square root of -1?”

    “What a fun thing to think about, Althea! What have you figured out so far?”

    “I know that when I square 1 I get 1, and that’s why the square root of 1 is 1. But when I square negative 1, I get 1 too, so shouldn’t the square root of 1 be negative 1 too? But how can it be two things?”

    “Hmm, that’s a strange one, isn’t it? I think there are too many ones in this for me to keep track of things. Let’s switch to 3.

    "I’m going to try to say what you said, but with 3 and 9. 3 squared is 9, so the square root of 9 is 3. But negative 3 squared is still 9, So why isn’t the square root of 9 equal to negative 3 also? Is that basically the same question you asked?”

    “Yes. Except the square root of 9 can’t have two answers. Can it?”

    “Well, somebody a long time ago decided that there should be just one answer for the square root of a number. And so we say that there is the square root of 9 and also the negative square root of 9.”

    Sunday, August 25, 2019

    Another Semester, Starting with Good Intentions

    My handouts are copied, the piles are organized on my desk. My rosters are printed. (I entered student names into excel, so I can organize things my way.) I've looked over my computer folders and found a few more things to share tomorrow. And I'm getting better at using Canvas' features - I plan to have students evaluate the new activities online, to help me decide whether each activity stays, goes, or gets improved.

    Will I manage to set up a new student survey in Canvas for each new activity?
    Will I blog about my classes, like I'd like to?
    Will I do more activities and less lecture in each class?

    Once the semester gets rolling, it's hard for me to change things up. It's so much easier to do what I have done before. May my passion keep me improving, all through the term.

    Saturday, August 17, 2019

    First Day, Once Again

    I've seen some great advice for the first day of class.  (Here's the one I've read more than once. I've seen other great ideas, but I don't see them now.) I'd summarize my take on this article as:
    • (Community) Start learning names, and get them learning each other's names.
    • (Expectations) Don't spend much time on the syllabus; there are more important things to do. (Have them read it, and then you can quiz them on it the next day, or just ask for their questions.)
    • (Learning, Expectations) If you use group activities (which are a very effective way to help groups of students learn), then you want to introduce students to this on day one.
    • (Curiosity) An activity that helps them see what's coming in the course would be especially nice.

    Our classes start in ten nine days. I am getting ready...


    First day activity for Calculus I 
    I have them graph a parabola (y=x2), then draw a tangent to it at x=3. (Some don't know what that means, so I walk around checking.) And then estimate its slope. After they're done, I get to talk about what makes actually finding the slope hard - such a good intro to the course. And they've had time to review graphing a parabola.


    First day activity for Linear Algebra 
    I have them solve a system of 3 equations in 3 variables. I  ask them to:
    • Write down a description of the process, 
    • Solve the system, 
    • Now figure how to check whether your answer is correct. (Naysayers, has the group done enough to be sure that the answer is correct?) 
    • Extra: What does each equation represent geometrically? What does the solution represent geometrically? 
    Once again, a great introduction to the themes of the course.


    First day activity for Geometry
    I taught geometry earlier this summer for the first time. I had them draw a triangle (and make sure it was different than their neighbor's). Find the midpoints of each side (they could measure or fold). Connect each midpoint to the opposite vertex. I hoped most would do it well enough that the 3 connecting lines would intersect at one point. My goals were to highlight: vocabulary, shapes, construction (which we were not doing with straightedge and compass - yet), conjecture, and the possibility of proof.



    First day activity for Precalculus
    I've been thinking for the last few weeks about what I'd like to do for Precalculus. I have found exciting activities in the past that turned out to be way too hard, and intimidated the students. I have a lovely fractions activity, but that doesn't represent what we'll do going forward.

    I am working hard to create an activity that looks at functions (and circles too) from 4 perspectives: equations, graphs, tables of values, and stories. I have 7 types of relationships (linear, quadratic, polynomial, rational, exponential, periodic, and circles). I don't have stories yet for the polynomial and rational. (My eternal gratitude to anyone who can give me a story I like for either of these.) And I won't show an equation for the periodic. (A trig function wouldn't make sense yet. But we'll get to discuss that.) So that makes 25 "clue sheets".

    I have 40 students, who I'll put in 10 groups of 4. Each group will start with two clue sheets. [So 5 of the sheets will not be handed out at first. I can label those as graphs on the back, keep them at the front, and let student turn them over once they're pretty sure they didn't find a graph match to their set.] Each will describe a different type of function/relation from one perspective and ask them to do a few things. Then they pair up with a clue sheet for each pair, and go looking for the matching clue sheets (same function/relation, different perspective). They go back to their group and explain to each other what they found. (I'll have extras up front, so anyone done early can work on a 3rd function/relation.) When we're done, we'll have a summary of the function types we'll be studying all semester.

    I dreamed some of this up late last night. When I started working today, I worried that it would be too hard. (I make up some crazy stuff sometimes when I'm falling asleep.) So my goal as I put this together has been to scaffold it enough. I am assuming some comfort with linear functions, and some familiarity (but not comfort) with quadratics and exponentials. They may not have encountered the others. (And most will not know any trig.)

    I put my first second draft into a google doc here. Your suggestions may help me improve it. (I decided to leave out the rational function. 6 functions with 4 clues each would be 24. One story and one equation are left out. That's 22. The last two clues will sit up front.)

    Edited (8/17): This is a great activity, but too complex for day one. I will do it on day two. On day one, we will review linear functions in a similar, but much simpler way. Here is my handout.




    My Goals:
    • Review plugging values in for x to find y. (Do they remember that b0=1? What else might trip them up?)
    • Review graphing.
    • See functions/relations in the context of modeling a situation.
    • Identify functions/relations by type.
    • See precalculus as a place to strengthen their understanding of all of this.

    If you use this activity, please let me know what changes you decide to make and how it goes.

    Monday, June 17, 2019

    Teaching Geometry - Journal Post #1

    I usually try to write blog posts with a point. I don't often have the energy for that lately. But I do want to keep track of how this class goes...

    Today was the first day of my geometry class. (It is the first time ever for me to teach geometry. I have done lots and lots of prep.) It goes for 6 weeks, and meets 3 1/2 hours a day. (Yikes!) My students are mostly high school students, though this is a college class. The college screwed up registration. I had 28 students on my list, but a bunch thought they were on a waitlist (only happens when a class is full with 40 students), and came in not yet registered. I now have 35 or 36 students. Of course I ran out of handouts.

    I gave what I thought were good directions for our first activity, and students managed to do strange other things.


    My directions:

    Welcome and Triangle Activity

    This activity is an introduction to the themes of the course. The more thought and care you put into it (the activity and the course), the more learning and joy you will get out of it.

    With a straightedge, draw a big triangle on this page. Make it as weird as you want as long as it is a triangle. Make sure it looks different than the triangle your neighbor draws.

    Now you are going to draw three lines inside it. Each line will go from a corner (aka vertex) to the middle of the opposite side (aka midpoint). These lines are called the medians of the triangle. How will you find the midpoints? Do this as accurately as you can. Now look at what others have drawn. What do you notice?

    [Can you guess how this activity connects with the themes of the course?]

    ***

    I should have had them find the midpoints first. And then talk about connecting those to opposite vertices. (Fixing the handout now.) Some people connected the midpoints to each other. Some drew a new triangle inside the first triangle. (Pretty but nothing like what I asked for.)

    So the activity took lots longer than I expected. Sadly, I didn't note the time. (I have a tutor working with me in class. I will ask her to keep track of how long things take.)

    I thought I could manage to lecture on 4 sections of our textbook in about an hour and a half. No way. I enjoyed what I did with the students, but we only got through 2 sections.

    Some of what I did was impromptu. I asked them to each write down their own definition of angle. Then I asked for volunteers to read their definitions. I talked about courage. I waited. Finally I got a volunteer. And then another. I did not criticize their definitions, even though I didn't like them much. The third one was closer. He was struggling for a word for how far the one side is from the other. I repeated his definition, with a big pause where he had been stuck. And then I said rotation. We talked about things that rotate.

    One of the book problems we did just asked them to measure a line segment against a ruler printed in the book, and angles on a protractor. It was hard for them. And that was good for me to see. It tells me where we are.

    I hated our textbook when I first started reading it, but eventually it seemed fine to me. I think the first chapter has more problems than the rest of the book.

    1.1 Reasoning. They talked about types of reasoning and included 'intuition' (along with inductive and deductive reasoning). No. Intuition helps us guess what we might want to try, and helps us see things that might be true. But I don't like them calling it one of 3 types of reasoning.

    1.2 on measurement turns out to be more useful than I expected. Looking it over now, there's a lot more there that I haven't gotten to. Good thing I've asked them to turn in their notes on their reading. I hope that helps. (They are supposed ot read the first 4 sections.)

    The last hour and a half is devoted to labtime. We will be using lots of labs from Henri Picciotto's Geometry Labs (free). I had hoped to do 3 of them today. We didn't even finish one. (Lab 1.1 involves putting different shapes of blocks around a point.) I had also hoped to introduce them to euclidthegame. We'll do that tomorrow.

    Quiz first thing tomorrow. I am hoping to keep them on top of things by quizzing daily.

    We are part of a program called High School STEM Connection. 20 of my 35 students are part of that program, and are required to come to tutoring after class. Most of them did. They got most of their homework done. Yay them. I love spending time with them where I'm not in charge. I'm just there to help out when they need me.

    I am hoping to complete 5 sections of the book tomorrow. I'm tempted to throw out the cool group activities, but I know I shouldn't. I think I'll just see how far we get, like today. And then tomorrow evening I'll have to think again about how to adjust.

    Sunday, May 19, 2019

    Teaching Basic Geometry




    Our college offers a course that is pretty much the same as high school geometry. I have never before taught it because we only offer it in the summer, and summer semester is squished into 6 weeks. I haven't wanted to teach math on that sort of schedule because it seems like it would be hard for  students to learn well that quickly.

    But I'm under load and I love geometry, so I will be teaching this course in a month. I've been preparing ever since January - reading the text, making plans and handouts, doing constructions. Geometry doesn't have the same flow for me that algebra does. I thought that might be because I've been teaching algebra for 30 years. But other math educators have also noticed how much harder geometry can be. Brain teasers everywhere. I like that, but I also need to figure out how to make this doable for the average student. I have gone through the book twice now, thinking about what they'll be learning, and hoping to notice the places they'll get stuck.

    We will be doing lots of compass and straightedge constructions. Even though I loved geometry in high school, I hated that compass. I'd press too hard and open it further in the middle of an attempted circle. Or not hard enough, and then the pencil wouldn't draw. Arghh. Computer tools make that a problem of the past. I've mentioned two sites in previous posts, sciencevsmagic.net and euclidea.xyz. But euclidea won't let you unlock a higher level until you succeed at the lower level, and I found that frustrating. Henri Picciotto pointed out that euclidthegame.com is much like euclidea, but doesn't require you to solve one level before doing the next.  After checking it out, I'm not sure I have any reason to go back to euclidea; euclidthegame is much more satisfying.


    Translating a Line Segment
    So one of the tools I'll be using with my class is euclidthegame. I had a blast working through the levels back in January, and wanted to redo them now to see which sorts of things might be hard for the students. I got stuck on level 7, translating a line segment. I replicated two solutions given in the comments, but I didn't understand why the steps worked.

    When I looked up 'translate a line segment', all of the sites I found said to hold your compass at the right size and move it over. That doesn't work on euclidthegame, and wasn't part of the original Greek compass and straightedge protocol. I almost gave up on understanding this construction, and thought about telling students they could skip this one, since our book does it the easy way too.

    People say believing you can do something helps you do it, but I often succeed right after giving up. Ornery, I guess. I doodled a bit, and realized that thinking about a parallelogram was the key. After that, I was able to solve it quickly. I want to share my solution here. (Turns out, this idea was in the comments at euclidthegame, I just hadn't seen it.)

    The idea is to make a line segment congruent and parallel to AB, with C as one endpoint of the new segment. If you imagine making a parallelogram with the old segment and the new one, and if you know that diagonals of a parallelogram intersect at their midpoints, it becomes relatively straightforward.
    1. Sketching parallelogram ABDC, 
    2. We know that we can construct the diagonal BC, 
    3. And find its midpoint (E). 
    4. Then make a ray from A through E, 
    5. And a circle centered at E through A. 
    6. Where that circle and ray meet will be D.  (Because E is now the midpoint of AD.)
    7. CD is the translated line segment we wanted.
     Lovely.

    The students won't have enough information when they get to this construction in the book (chapter 1) to do it the right way. We'll have to wait until chapter 4, where they will find out why those diagonals bisect each other. I sure am glad to know that ahead of time.


    Planning the Course
    We will be meeting from 8 to 11:35 am four days a week. Who is going to get there on time at 8am if they don't have to? So I will start with a quiz every day. If they don't do well, they can retake it (new questions, same topics) outside of class time. [So a real sleepyhead could still ace the quizzes, even if they came late. But that will take some serious dedication. That's cool.]

    Then I'll do my usual combination of lecture, pairwork, groupwork, and guided discussion from 8:15 to 9:45. After they take a break they will have lab time. Partly because I don't believe anyone can concentrate for 3 1/2 hours in a regular classroom, and partly because I think they really need to get their hands on some of these tools. I'll have geoboards and dot paper to record their geoboard results. I'll be using most of Henri Picciotto's lovely Geometry Labs book. (Free. Thank you, Henri!) And they'll be using both euclidthegame and geogebra to do their constructions. Lab projects are 20% of their grade. They'll have to do constructions, geoboard activities, proofs, and one activity of their choice.

    Anyone teaching geometry who'd like more details, please ask. And anyone who would like to share tips, I'm all ears.




    Monday, April 1, 2019

    Playful Math Ed Blog Carnival (aka Math Teachers at Play) #126

    126 = 6*21 = 2*3*3*7


    If you want to choose 4 chicks randomly from 9 total chicks, there are 126 ways to do it.


    Students learn more if they make up the stories for story problems themselves. Can your students make up stories for these ways of making 126?

    126 = 27 - 21 (difference of powers of 2)
    126  = 42 + 52 + 62 + 72 (sum of consecutive squares) 
    126 = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 (sum of consecutive even numbers)


    This blog carnival has evolved from being mainly contributions to being mainly items the blog host has discovered. Since my passion lately is geometry, this issue is dedicated to geometry. (Which of the 3 ways of making 126 above has a geometric interpretation? Hint: There's a picture of it here... somewhere...)



    Constructions
    I have been intrigued for the past few years with Archimedes' method of finding pi. He didn't have the square root symbol, so he approximated using fractions, getting pi between 3 10/71 and 3 1/7. But if we follow his steps, and keep the square roots, we get a lovely pattern for our answer.You can try it. Construct a hexagon in a circle. If the radius of the circle is 1, then the hexagon's perimeter is 6. Perimeter over diameter = 6/2 = 3. Now create a dodecagon (12-sided polygon) from the hexagon. You can find the side lengths from repeated use of the Pythagorean theorem, and then find perimeter over diameter. Your result will be closer than for the hexagon. You can repeat this process until a pattern emerges.


    If you want to get better at geometric construction (straightedge and compass style), play with it at sciencevsmagic or euclidthegame.

    You can improve your geometric reasoning skills with the puzzles in Geometry Snacks (and More Geometry Snacks), by Ed Southall and Vincent Pantaloni. There are more puzzles at his blog. If you like them, the book is a treasure trove.

    Because I've fallen in love with geometry, I decided to teach it this summer, for the first time ever. So I'm doing a lot to prepare. Henri Picciotto is an expert geometry teacher who graciously offered me his time over breakfast. He advised me to download his Geometry Labs book (free) from his Math Ed Page site. There is so much more there than this. But this alone was a huge gift. I think it may transform my course.


    I've been collecting geometry mysteries. Medians are the lines from midpoints of the sides of a triangle to the opposite vertices. The 3 medians seem to always cross at one point. Why is that? I tried for weeks to prove it, and just couldn't. I finally gave up and looked at the proof. (And told my students how much fun I had failing!)   I then found another proof that followed a very different path. Can you prove it?

    Here's a simpler mystery: If you make a 5-pointed star (perfectly even, I can't do that without digital help...), what is the angle at each point?


    One of my favorites for seeing the geometry in math topics you didn't know were geometric is  Magic Pi - math animations. I hate that they're only on Facebook because I am not comfortable linking to facebook in class. But they are amazing. (I linked to one that's pure geometry. So cool.) They apparently do most of their animations in geogebra. I am a complete novice next to them. Here's a geogebra sketch I made today. It might be my first in their 3D mode.





    Making Your Own Math
    At the beginning, I mentioned having students make up their own story problems.  Here's a lovely post from Arithmophobia No More about just that. Here's another angle on teaching story problems, from Jen at Math State of mind. Leaving out the numbers helps students to slow down.

    This blog post, by Amy at When Life Gave Us Lemons, is about her son making up his own math games. And John Golden has a whole class make up variations on a game he shared with them.


    Denise Gaskins, founder of this carnival, pulls together so many books and ideas I love in this post. I don't know how she does it! The (surface) topic is fractions, but more than that, it made me think about how we can help students learn by saying less. The video she includes, with a teacher asking the two boys questions, and never telling them they're wrong, is fabulous. One of the commenters at Denise's post linked to a discussion of his own with a student. And that made me think about Bob Kaplan's guide to 'becoming invisible' (or not giving away the math). (What math delights have you found lately by following your nose? Bunny hops rock!)



    You can check out the Carnival of Mathematics here. And if you'd like to host this carnival (we need help next month!), you can learn more and sign up here.



     

    Friday, March 22, 2019

    Coming soon: Math Teachers at Play (aka Playful Math Ed) blog carnival

    I'll be posting the blog carnival here sometime late next week. Right now I'm beginning to gather links to lovely, playful math ed posts (and sites and videos and ...). If you know of something I should include, please email me at mathanthologyeditor on gmail.



     
    Math Blog Directory