Hurricane Katrina 2.0? (From Baton Rouge with Pictures)

If the United States ever gets hit with a major disaster, there will be at least one group of people who will know what to do immediately: the folk from southern Louisiana.  Within hours of such an event the volunteer Cajun Navy would have already begun rescuing victims, Louisiana churches and civic groups would have begun serving hot meals, the Celtic Media Center would have provided shelter using their movie sets, and LSU would have created a medical command center for nursing the sick and wounded back to health.(*)

Learning Division Facts: Numbers that are Products of Two 1-digit Numbers

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 7-year-old Autumn tells whether a number from 1-100 is the product of two 1-digit numbers.  This is an important step along the way of using the multiplication table to learn the division facts.

Mathematician Pallavi Dani – Divergence in Right-angled Coxeter Groups

Intended Audience: Everyone, and especially teachers who want to show to their students a mathematician explaining the motivation behind their own research.

In this episode we meet Pallavi Dani, a mathematician here at Louisiana State University, who talks to us about using geometry to study problems in algebra and vice versa.

But Honey, You do know the Multiplication Table!

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 7-year-old Autumn realizes that she does know the complete multiplication table!  Here’s the story:

Fraction Multiplication via Skip Counting!

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 7-year-old Autumn uses skip counting to find $7 \times \frac{2}{3}$ as well as $4 \frac{4}{5}\div\frac{2}{5}$ in her head!

A Portrait of an Inspiring Teacher: Fred Reusch

Fred Reusch: Teacher, Tutor, ‘Coolest Man Ever’

You may remember my dad from one of my earlier posts, What it is like to be the son of Captain Derivative, and his picture:

Here are a couple of my thoughts about Rademacher’s article and my dad that will hopefully add a little “lagniappe” to the piece and Fred’s teaching:

Posted in Commentary, That's News To Me | | 1 Comment

Clayton Shonkwiler: Research Seminar on Applications of Geometry and Topology to Random Walks

In this episode we see a research talk presented by Clayton Shonkwiler, a mathematician at Colorado State University, who is speaking to us about applications of geometry and topology to random walks/polygons and polymer physics.

Can an atom have two nuclei? An inquisitive child wants to know!

My 7-year-old daughter, Autumn, and me were discussing how an atom is made up of mostly empty space with a small nucleus of protons and neutrons in the “center” and electrons swarming about it.  During the discussion, she asked an insightful question,

“Can an atom have two cores (two nuclei)?”

The Washington Post: Many parents hated Common Core math at first, before figuring it out

Columnist Jay Mathews writes in a new Washington Post article, Many parents hated Common Core math at first, before figuring it out,

Dear Mr. Mathews: Thank you for sharing these stories. Continue reading

Mathematician Clayton Shonkwiler: An Advanced Perspective

Intended Audience: Mathematicians, graduate students and ambitious high school students.

We continue our interview with Clayton Shonkwiler on applications of geometry and topology to random walks/polygons and polymer science, but now at a graduate student level. To see the earlier interview with Clayton geared at a high-school level, go to: Mathematician Clayton Shonkwiler talks about Polymer Science.

Growing up with Eureka Live: Mathematical reasons for introducing a different solution method

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 7 year old Autumn leads an audience of about 100 teachers in doing arithmetic problems.  Watch her explain how she uses different methods to solve problems other than the “standard column math” method (algorithms), and allow me to explain why it is important to build curricula that encourage all of these methods.

Holiday Math Special: In the 12 Days of Christmas song, which of the presents do you get the most?

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

Recently Autumn asked, “In the twelve days of Christmas, which of the presents do you get the most?”  This is the type of question you hope your child asks you, because it can lead everyone in the family on a great adventure where math just “happens” in the course of thinking through the answer.

Multiplying by 25

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

Autumn is now 7 years old! Thank you to everyone out there who has been watching her grow up.  We have many more great videos planned for this year, so please like us on Facebook (www.fb.com/ScottJBaldridge) or follow us @ScottBaldridge, or follow this blog for updates.

Woman Superhero T-Shirt Challenge

Autumn asked, “Why don’t you wear a girl superhero T-shirt for our videos, dad?”  And I said, “Why not?  Absolutely!”  We started searching for a woman superhero T-shirt for men but ran into trouble:  There are a lot of women superhero T-shirts for women, but very few for men that are perfect for our video series.  So we need your help!

If you know of an excellent woman superhero T-shirt for men, please share it with us in the comment section below.  If you include your name (you don’t have to), we will thank you in the video when I wear your T-shirt!

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge

Mathematician Clayton Shonkwiler talks about Polymer Science

Intended Audience: Everyone, and especially teachers who want to show to their students a mathematician explaining the motivation behind their own research.

In this episode we meet Clayton Shonkwiler, a mathematician from Colorado State University, who talks to us about applications of geometry and topology to the study of random polygons and polymer science. Continue reading

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn explains that a variable is a slot that you can put a number into.  The slot is usually represented on paper as a letter (such as x) or a mark (such as ___).  Here’s the definition of a variable symbol: Continue reading

Building the ⟨Sci|State⟩ Studio—a weekend in time lapse.

The SciState Studio is finished and we’re all ready for production! In this video, we get a behind the scenes look at some of the finer details of our studio construction. As we embarked on this adventure, one of the first things we learned was that high-quality audio is an essential component of any successful video series. With this in mind, we have outfitted our studio with acoustic insulation panels and installed various other echo reducing components throughout. We also constructed an awesome plexiglass “light board,” which is similar to a traditional whiteboard but doesn’t require one’s back to face the camera. We’re really proud of everything we’ve produced and hope you agree!

We should point out that it takes a real expert (thank you Justin Reusch!) to set up the lighting for the video stage.  You can see how that came out by watching our “Testing out the Green Screen” video.  As a point of comparison, note the terrible sound in the “Testing out the Green Screen” video—that is what the studio sounded like before we built the acoustic insulation panels.

As always, comments are welcome below.

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708

Something fun: What it is like to be the son of Captain Derivative

Above is a picture of my stepdad, Captain Derivative, doing integration exercises for yet another day in the vector field against his arc enemy, the pathological and degenerate Prime Matrix.

Some people have asked me, “What is it like being the son of a differential operator who can wield power series with such ease?” Well, when I was young and still very near my initial value, I used to oscillate rapidly between believing whether his exponential powers were real or imaginary–often shifting my phase over time with some frequency. But then he removed a discontinuity from a complex surface right before my very eyes, and I saw the proof of his limitless analysis: I converged upon the realization that my own scientific skepticism had been irrational. From that moment on (t=5), I would often help him from his secret power base by relaying coordinates of tangents he could approach in the Cartesian plane so he could do battle against those divergent improper integrals.

Go Captain Derivative, we are with you (or at least, within an epsilon ball of you). May the functions you differentiate always be smooth!

(My stepfather, Fred Reusch, is the calculus teacher at my old high school in Rockford, Michigan.  Today was “Super-hero Day” at the school, and he came dressed as “Captain Derivative.”  I think he would wear this outfit to school everyday if he could!🙂 )

CHANNEL: That’s News To Me

Posted in That's News To Me | | 1 Comment

Adam Saltz speaks on an annular refinement of the transverse element in Khovanov homology

Intended Audience: Research mathematicians, professors of mathematics, graduate students in mathematics, and advanced undergraduate students in mathematics.

In this 1 hour episode, we see a presentation by Adam Saltz, a mathematician and graduate student at Boston College, on a new invariant of transverse knots in links coming from Khovanov homology.

In the talk, Adam discusses some of the details contained in his paper with Diana Hubbard, An annular refinement of the transverse element in Khovanov homology. Here is the abstract to their paper:

We construct a braid conjugacy class invariant κ by refining Plamenevskaya’s transverse element ψ in Khovanov homology via the annular grading. While κ is not an invariant of transverse links, it distinguishes some braids whose closures share the same classical invariants but are not transversely isotopic. Using κ we construct an obstruction to negative destabilization (stronger than ψ) and a solution to the word problem in braid groups. Also, κ is a lower bound on the length of the spectral sequence from annular Khovanov homology to Khovanov homology, and we obtain concrete examples in which this spectral sequence does not collapse immediately. In addition, we study these constructions in reduced Khovanov homology and illustrate that the two reduced versions are fundamentally different with respect to the annular filtration.

This video and paper are aimed at mathematicians, graduate students and undergraduates with lots of experience in topology.  However, high school students who think they are potential math geniuses may still enjoy looking at it to see what advanced theorems and proofs look like.

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708

Testing out the new Green Screen (Plus a bonus 14×16 calculation)

This is a test video that Shea and I did to test the new green screen. Shea is spoofing a bit with me—acting all surprise at my ability to calculate 14×16 in my head. He looses it slightly near the end.  (Research mathematicians do a lot more than mental math although we are almost constantly doing mental math: it is just math that is a lot harder than simple arithmetic.) Check out Shea laughing at the end.

The test video is to try the different types of backgrounds we can now use with the new studio setup (and green screen). You can vote which background you like the most here:

This is just an informal poll.  You can comment below if you have a better idea for a background.  For example, Shea’s son wanted whales swimming.  Maybe not quite professional enough for math videos…

Thanks especially goes to Justin Reusch, who came all the way from Austin, Texas to set up—and explain how to set up—the lighting for this type of shoot. You can see his work in the video on our faces (the back-light halo effect on our heads, the side lights and shadows on our faces, etc. ). Thank you, Justin!  Thanks also goes to Hang, who kept the kids busy while we put together the shot.  You can hear the kids in the background playing during the video: like I said, this was just a test video.

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636 and NSF grant DMS-1249708

Powers of 2

Intended Audience:  Teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn explains that the meaning of “2 to the power of 4” is a product of four factors of 2:

$2^4=2\cdot 2 \cdot 2\cdot 2.$

This is the beginning of exponentiation and it is easy to understand—the power (in this case, 4) tells us how many factors of the base (2) there should be.  I usually tell my college students tongue-in-cheek that, “Mathematicians are laaaaazzzzzy. We came up with the notation $2^{30}$ because we got tired of writing all 30 factors of two out!”

There are a few issues to watch out for when introducing a child to exponents (some of which I mentioned in the video):

• At this level, after students understand multiplication, powers of 2 are no harder or easier than learning addition or multiplication facts.
• However, it is a new operation and one that is easily confused with multiplication.  The statement, “2 times 4,” sounds a lot like, “2 to the 4th.”  Furthermore, the processes for evaluating the expressions are are similar but for powers we use multiplication instead of addition: “2 times 4” is describing a number of addends while “2 to the 4th” is describing a number of factors.

For these reasons, you (as the parent or teacher) should be very deliberate about asking, “What is 2 to the power of 4?” to help your child/students understand that you are asking for something very different than “2 fours.”

• Definitely stick with “powers of 2” until students are comfortable with the meaning of exponents and can confidently tell you what $2^1, 2^2, \ldots, 2^{10}$ are.  Remember, you are trying to help students learn the meaning of the operation, not memorize a bunch of numbers. (If you feel you can move on, go to powers of 10 next, and then to powers of 3.)

Once they are comfortable moving back and forth between $2^7$ and $128$ the real fun can begin—learning the properties of exponentiation like $a^m\cdot a^n=a^{m+n}.$  Don’t worry about this for now; Autumn and I will show you how easy and pleasurable it is to learn some of these properties in another video.  For now, just concentrate on learning the powers of 2 themselves (up to 10 or 12).

• Finally, it is much easier to remember the powers of 2 if you use “tags” or “pins,” i.e., memorizing a couple of easy-to-remember powers of 2, and using them to quickly figure out the rest.  I recommend: $2^5=32$, $2^8=256$, and $2^{10}=1024$.  Then one can quickly find $2^{11}=2^{10}\cdot 2= 1024\cdot 2=2048$, which is just an easy double.  You will see the “story” Autumn and I used to remember $2^8=256$ in the video.

Check back soon for a new video on the properties of exponentiation.  In the meantime, enjoy learning the powers of 2 with your child/students!

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

An article on how Eureka Math began.

Leigh Guildry wrote a nice piece for The Town Talk newspaper, “How did Eureka Math start? 2 writers answer.”  She traces the steps that I, Robin Ramos, Nell McAnnelly, and Lynne Munson (director of Great Minds) took to develop the Eureka Math/Engage NY curriculum.

She starts with recent test results of students from Rapides Parish School System who were using Eureka Math:

“Rapides [Parish School System] students progressed at almost double the national average on benchmark tests by Discovery Education. Students gained an average of 141 scale score points during the 2014–15 school year, compared to 73 points for students in districts nationally, according to eureka-math.org. The district more than doubled the average U.S. gains in middle school grades (6–8) and in kindergarten.”

Pretty impressive! Congratulations go to students and their teachers.

The article goes on to interview Pam Goodner, who was the lead writer who worked with me in creating the 12th grade “Precalculus Course.”  (Thank you, Pam, for your hard work.)

Louisiana Bayou, JR Meeker (1884) wiki commons

CHANNEL: That’s News to Me

Electric Girls: Shaping Role Models in Technology

Check out the video below, then visit their website to learn more about the program.  I need to see if there is a similar program in Baton Rouge for my daughter Autumn.

From the website:

“Electric Girls is a 12-week education program for girls ages 9-14 in New Orleans. Using a mentorship structure, we teach girls to become leaders and role models in STEM (Science, Technology, Engineering, and Math). Girls come away with a new set of hard skills (soldering, drilling, building circuits, etc.) and soft skills (perseverance, curiosity, leadership, self-motivation).”

Watch for Maya Ramos in the video.  (She is Robin Ramos’ daughter.  Robin is a good friend and colleague, and the lead writer/teacher of A Story of Units.)  Maya studies music in New Orleans when she isn’t building electrical circuits.  You can listen to Maya’s piano playing in her band “Spare Change” by visiting their Facebook page.

CHANNEL: That’s News to Me

Posted in That's News To Me | Tagged | 2 Comments

The Secrets of my Teaching Success: James Tanton

Note from Scott Baldridge:  I recently asked James Tanton, “What are some of the reasons for your success as a teacher?”  I consider James to be a fabulous high school teacher and a good friend, and was interested in his response.  To see why so many people think highly of James, I recommend that you visit James’ websites www.JamesTanton.com and www.gdaymath.com and follow him on twitter: @JamesTanton (and click his suggested links below).  James responded with the following 12 points. I felt they were so well-articulated and compelling that I couldn’t let these gems remain just between the two of us.  Thank you, James!

The Secrets of my Teaching Success

by James Tanton

What are some of the reasons for your success as a teacher?

And I sat down recently to try to give a serious answer to this question.

But I first need to point out that I disagree with the underlying premise of this question: I am not sure that my success means that I am a “good” math teacher by the usual standards of teaching. People may be shocked to learn that I always arrange the tables in my classes and workshops so that everyone is facing the board. I lecture, I only lecture, and all is focused on the teacher in the room. I use the board extensively, and there is absolutely no technology anywhere in sight as I teach. And I don’t do anything innovative in the classroom – seriously, zero, zip, zilch on the innovation front.

So this ego-full, self-focused piece is my attempt to answer the question as to why some people seem to think I am a successful teacher despite the above. It comes as twelve points.

Success 1: I have a cute accent.

I am serious in that I think this a big part of my success in the classroom.

I was raised in Australia with a British father and as a result my accent is something confusing: Australians think I am British, Brits know I am Australian, and everyone else is confused as to what I am. But my accent seems to be extremely pleasing to the American ear and I am fully aware that it works to my full advantage in my American life.

Success 2: I treat everyone like adults – even kids.

I always assume everyone just does the right thing.

Well, I need to qualify that. We are all human and I know we make silly mistakes when under stress and pressure and so might slip on doing the right thing every now and then. But that is the learning process for all this.

So I assume people just do the right thing, and if they don’t, will learn from the goof and just not do it again.

I had one blatant, silly act of cheating as a college professor: I received two identical, word for word, silly error for silly error, homework papers. My response was to give one paper an A+ and the other a C- and never say a word. It turned out that the two authors never said a word either and it never happened again.

I once gave a lecture on cheating 101, general pieces of basic advice on how to get away with things. (If you’ve copied someone’s paper, don’t hand it at the same time as your partner-in-crime: make sure your papers sit in different parts of the pile. Photocopying someone’s answers is just a ludicrous idea. Don’t copy the same spelling and obvious math mistakes – perhaps insert a few more of your own. Don’t wear a baseball cap during an exam – the rim points the same way you are looking. And so on.)

When I moved to high-school teaching I was flabbergasted at the idea of “needing to remove the temptation of cheating” for our students. This incorporated ideas such as erecting screens between seats during class quizzes and having students being supervised while they do make-up tests. Where are students meant to learn about the wrongs of cheating and making those first-time silly mistakes? Plus the insult to students assuming they can’t be adult about all this! I did none of the things I deemed insulting to students.

There was one time when young Jenna was looking over at someone’s paper during a quiz. I just walked up behind her and quietly whispered: “Just be careful where your eyes go during a quiz.” End of issue. I just ask students “to do the right thing” when it comes to finishing up a test at home. Even if they slip, there is an emotion that accompanies the wrong doing that sits and lingers, and contending with that emotion is the learning experience. (Plus students, by and large, do do the right thing!)

Success 3: I am quirky and I like to play with ideas.

Here’s a tiny piece of quirkiness that illustrates the power of playfulness:

In learning about permutations we start by counting the number of ways to rearrange letters in words or, better yet, in our names: the letters of JIM can be arranged 6 ways (3!), the letters of JAMES 120 ways (6!). But a name like BOB or DANA represents a problem. (Brute force gives 3 and 12 ways, respectively.) The problem is worse for the word CHEESE. So we need to figure out a reasonable way to handle repeated letters.

As ideas develop we go from CHEESE to CHEESES to CHEESIEST, and when we have the hang of it, we go straight to CHEESIESTESSNESS, the quality of being the cheesiest of all the cheeses. People just seem to love the “word” cheesiestessness and the whole lesson sticks.

I also love squine and cosquine (www.jamestanton.com/?p=605). I love to ask how many degrees there are in a Martian circle (www.jamestanton.com/?p=633).  And I love quirky words from the history of math: vinculumobelusradix, and so on (www.jamestanton.com/?p=1258).

Success 4: I think hard about “what’s really going on” and “why anyone cares.”

I think I am good at thinking deeply about stuff and can cut through all the usual surrounding clutter. That’s why my lecture style works, I think: what I ramble on about is de-cluttered content and so sustains interest.

Plus I do the quirky, straight to the heart-of-the-matter, lectures. Exploding Dots is a prime example. (www.gdaymath.com/courses/exploding-dots/)

Success 5: I break every 37 ½ minutes.

I once read a paper early in my career that said that the average attention span of an adult or near-adult audience member sitting through a lecture-style presentation is 37½ minutes. I’ve taken that as a literal fact, and have made it a universal law in my teaching. I tell this little story at the start of my courses and workshops and we religiously have a break at the 37½ minute mark, even if it is only a 45 minute class!

Success 6: I know some history of math.

I want math to be the human story that it is. I share the tales of the backs and forths and the struggles of developing ideas leading to how we see and use them today.

Success 7: I am not at all afraid to make mistakes. Even whopper of ones.

It is a vital and genuine part of math to be human in your relationship with it. I don’t need to be seen as the expert. But I do need to model what it means to engage with mathematics as a human being.

Success 8: I seem to be good at helping people feel it is okay not to know.

After all, I know very little myself. The message I give is that it is completely okay not to know something, but it is not okay not to want to find out.

This notion is tied into the use of the word should, as in “you should know this” or “students should know.” Should statements often have a feeling of judgement attached to them and they induce unpleasant sinking feelings in the gut for the recipients. I avoid making should comments.

And usually these statements are moot: even if students should know how to distribute a negative sign by grade 9 and your students don’t, it is irrelevant – it just means that you need to talk about distributing the negative sign with that class. (Try something like 1.4 of www.gdaymath.com/courses/astounding-power-of-area/.)

But there is another aspect of these “should” comments that worries me. As one’s mathematical sophistication grows one starts to see former concepts in a new light. Subtleties and hidden assumptions become clear and previously comfortable topics become uncomfortable and shaky. The idea that, for example, by the end of middle school students should be comfortable with fractions is ludicrous to me. Fractions are actually very hard and a thinking high-school student really should revisit them and be uncomfortable with them! (www.jamestanton.com/?p=1461.) (Did I just use the word “should”?)

Success 9: I think I am good at recognizing “hazy” thinking.

You know when you are lecturing or teaching on content that you really haven’t quite properly sorted out for yourself. You can do the work, you can explain the piece, but you know you don’t really “get it,” the heart of it, that is. I have lectured while in this state too, it happens, but I share my emotional state with the audience. I like to think it helps students recognize hazy thinking when it happens to them too. Hazy thinking is a call to go for a walk, to mull on the idea, and to ask “What’s really going on with this topic?”

Success 11: I have a PhD from Princeton of all bleedin’ places!

People seem to think that means I know my stuff. Hmm. That perception certainly contributes to my teaching success.

Success 12: I am not obsessed about assessment.

I just want students to prove to me that they get it in the end. If it takes a while before they do and grades are lousy during that period, no worries, get it in the end we can ignore all that. This notion seems to be an anathema in high-school world – of all places!

(If I am forced to think about assessment in high-school teaching, I think this way: http://www.jamestanton.com/?p=968.)

As always, please feel free to comment below!

CHANNEL: Engineering School Mathematics

Posted in Engineering School Mathematics | | 4 Comments

Something fun: Have you ever used a ruler to measure a Snafoose?

Have you ever used a ruler to measure a snafoose?
I’ve never, never-ever, tried to measure a snafoose.
Certainly not on a goose.
Or near a boar on the loose.
No, I never tried to measure a snafoose.

But I have used a ruler, why, I have used it a lot!
I have used it to find the distance between this point and thot!
Between two and three,
or three and eight, as I was toght,
to find the distance between four-thirty-three-point-four and two-point-two to the naught.

I’ve measured here and there, in good days and bad,
I’ve measured while eating,
I’ve measured while preening,
why I’ve even measured while dancing the twirl-e-bop-de-careening.

And here’s what I’ve learned, if you permit me to spin:
the distance between naught and three is just three again.
so maybe it should be no surprise:
it is also the same to the negation of three!

Absolute value does not need to be an absolute bore,
It just comes from measuring, measuring and measuring some more.
And when it’s finally brought up, in algebra, with letters,
kids who’ve spent time with rulers,
will best even those…who should know better.

— Scott Baldridge, 2007

I wrote this silly little poem in 2007.  (I can safely claim to be a non-poet.)  I stumbled across it in an email recently looking for another email and thought I would share it.  Note that even then, years before there was anything called “Common Core,” I was advocating that elementary students use rulers, protractors, beakers, weighing scales, etc. to build an intuitive understanding of units.  At the time I wrote this little poem, I did not realize that a few short years later I would be writing an entire curriculum, A Story of Units, based upon manipulating units.

(Check back later for a picture of a Snafoose that Autumn is designing!)

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

What is the sum 1+2+3+4+…+24+25?

Intended Audience:  Math teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn shows how to find the sum

1+2+3+4+…+24+25.

This problem actually has a rich history involving one of the greatest mathematicians who ever lived: Carl Friedrich Gauss. Please share Gauss’ fascinating story with your children or students. The story goes like this:  When Gauss was seven, his elementary instructor gave what the teacher thought was an extra-pointless exercise of adding up the first (say) 25 numbers just to keep his students busy for an hour.  Almost immediately, young Gauss threw down his slate and declared, “There it lies.”  When the hour was up, the teacher inspected Gauss’ answer and found it to be correct.  An interesting article investigating the full story can be found here.

Autumn’s solution to this question was probably very similar to how Gauss solved it.  In the video, I mention that Autumn ran up stairs and came down 10 minutes later with the solution.  Here is her actual work (click Autumns_Work for a .pdf version):

In this picture you can see her thinking: Pairing up 1+25, 2+24, 3+23, etc. and keeping track of which terms she summed on the line below (you can also see the left over 13 at the end).  I think what happened next is that she recognized (in her mind) that

$26\times 12 + 13 = (25\times 12 + 12) +13 = 25\times 12 +25 = 13\times 25,$

because she writes “same as 13 x 25.”  Regardless, you can see her calculating the

$25 \times 10 = 250$

just as she did in the video, and you can see me checking her work underneath (and actually showing her another mental math technique–look!).

For Teachers who teach Eureka Math:  This sum is called a finite series because it is a sum of a finite sequence of numbers.  We explore this particular series in many different places in the high school curriculum, but a particularly interesting discussion around this series occurs in Lesson 8 of Module 3 of Grade 9 (Algebra I).  In that discussion we show how the sum can be visualized as “triangles:”

–Grade 9, Module 3, Lesson 8 of Eureka Math/EngageNY

This visualization quickly leads (through pictures!) to the general formula of the sum of the first n positive integers:

$S(n)=\frac{n(n+1)}{2}.$

Plugging 25 in for n shows that $S(25)=25\times 13$, which is what Autumn derived.  If you are teaching Lesson 8 (or any of the other lessons where this series shows up), think about sharing this video of Autumn with your class as a way to stimulate a discussion with your students.

As always, comments are welcomed!  In particular, we are trying out different places to do math.  The math in this video was done on the hood of a Porsche 914.  We have friends with all kinds of cool sports cars who may be willing to let us borrow them for an afternoon, so look for more “Math on the hood of sports cars” soon (especially if we get positive feedback to do more videos like this).

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

Eureka Math Curriculum: A Breakout Hit

In his opinion article at US.News & World Report, Common Core’s First Breakout HitRobert Pondiscio discusses how the EngageNY curriculum (both the English Language Arts and Mathematics curricula) is being warmly received by school districts and states all across the country:

“I recently obtained data from the New York State Education Department showing that while EngageNY units, lessons and curriculum modules have been downloaded nearly 20 million times as of early May, more than half of those users have been outside of New York. EngageNY may be quietly emerging as Common Core’s first breakout hit.”

Why might Eureka Math/EngageNY be a breakout hit?  There are, of course, a number of reasons.  I hope and think that one of the reasons is because teachers have been searching for and finally found a mathematics curriculum that actually works for them–that they are seeing marked improvement in their students’ understanding of mathematics like no other curriculum before, and that news is spreading across the country.

I think teachers are the real breakout hits.

CHANNEL:  That’s News to Me

In Memory of Dr. Dolores Margaret Richard Spikes

Mathematics is…mathematics.  It may seem cold and impersonal at times—appearing not to be driven by human wants or desires but by the precise statements of assumptions and propositions that lead to the proofs of theorems.  This is just not so! Math is, after all, as human as art.  And yet, the seemingly impersonal nature of the discipline is actually one of its greatest assets.  Mathematicians’ shared intensity to irrefutable argument is one of the reasons why we as a group love to celebrate the beautiful mind of anyone who discovers and proves a new theorem.

Today I wish to celebrate the mathematical genius of Dr. Dolores Margaret Richard Spikes, a Ph.D. alumna of Louisiana State University who died last week.  Help celebrate one of her mathematical accomplishments with me: take a moment to read and absorb the abstract to her 1971 Ph.D. thesis:

Title: Semi-Valuations and Groups of Divisibility

From the abstract:  This paper gives procedures for constructing a class of groups of divisibility of rings (not necessarily domains) which properly includes the class constructed by Ohm.  Toward that end, we first extend the concept of a semi-valuation of a field to rings which may contain zero-divisiors.  The notion of a composite of two valuations of fields is then extended to the notion of a composite of two semi-valuations of total quotient rings (which may not be fields), and the construction of this composite is then related to an exact sequence of semi-value groups.  Necessary and sufficient conditions for this sequence to be lexicographically exact are given.

Dr. Spikes also made history in 1971 by becoming the first black graduate to receive a doctorate in mathematics from Louisiana State University.  She was only the 19th African American woman to earn a Ph.D. in mathematics at any university.  Her thesis work stands on its own and is a true testament to her intellect, but the circumstances of the times surrounding her achievement makes it all the more remarkable.  Thank you for your contributions to mathematics, Dr. Spikes.

I highly encourage everyone to read more about Dr. Spike’s rich and impactful life by visiting her Wikipedia page and reading the excellent reference articles there.  Also, check this out.

(photo: SUNO)

CHANNEL: That’s News to Me

A Quick Comparison of a State Assessment and Eureka Math

by Scott Baldridge and Ben McCarty

Intended Audience: Parents, teachers, and other educators involved in the Eureka Math/EngageNY Curriculum.

In this article, we simply discuss types of math problems from a state assessment and then show similar problems from the Eureka Math curriculum.  We focus on the new TCAP Achievement Test (called TNReady Math), mainly because one of the authors of this article (Ben McCarty) is an assistant professor of mathematics at the University of Memphis.  The format of this post is simple: we will discuss features of the TCAP and then show examples that match those features in the Grade 3 Eureka Math curriculum.

Example 1: Language

From the old Grade 3 TCAP sample test:

and a similar problem from Eureka Math Grade 3 (Module 1, Lesson 13):

Note the similarity in the language of “number sentence” between the two problems. Language like this will very likely continue to be used in the new TNReady assessments. Students using Eureka Math will be prepared for that language.  This aspect is particularly important for fractions in 3rd and 4th grade where Eureka Math uses the exact same language (like “unit fraction”) as specified in the Tennessee Academic Standards.

Example 2: Explain-Your-Reasoning Problems

As explained in the “Seven Things You Need to Know About TNReady Math” document, the new TNReady assessments will no longer be all multiple-choice and instead will have a variety of formats:

3.     TNReady will replace the state’s multiple choice only test in math and will include a variety of questions.

Eureka Math teaches a variety of answer formats that should mesh well with the types of questions asked on TNReady.   Eureka Math also gives insight and guidance to the teacher in how to model ways to answer these problems (cf. the teacher-student vignette on the left hand side):

This problem was taken from Grade 3, Module 3, Lesson 11.  This one explanation describes three different models/strategies that students can use in explaining their answer: (1) The tape diagram at the top, (2) the use of letters to represent unknowns, and (3) several mental math strategies for finding 72-28.  Taken altogether, and enacted in daily use over the entire year, these explanation strategies become part of the bread-and-butter methods students can use to answer TNReady assessment problems.

Example 3: Multi-step Problems

Also explained in the “Seven Things You Need to Know About TNReady Math” document, the new TNReady assessments will have many more multistep problems than the old assessments:

4.     TNReady will ask students to solve multi-step problems, many without using a calculator, to show what they know.

Multistep problems are one of the prominent features of the Eureka Math curriculum; it truly exceeds other curricula in preparing students for multistep problems.  The problem in Example 2 above is one such example, but for good measure, here is another example:

The tape diagram (the picture between the two paragraphs) helps students convert this multistep problem into pictures they can use to solve the problem.  The TNReady assessments will expect students to be fluent with answering word problems using tape diagrams and other models because they are part of the standards at every grade.  Compared to the mostly-one-step problems of the old TCAP, students using the Eureka Math curriculum will be ready to excel on the new TNReady assessments.

Scott Baldridge
Distinguished Professor of Mathematics,
Louisiana State University
ScottBaldridge.net (This article and other Engineering School Mathematics articles can be found at this website)

Ben McCarty
Assistant Professor Mathematics
University of Memphis
Mathematician, PK-5, EngageNY Mathematics Curriculum
http://umdrive.memphis.edu/bmmccrt1/public/

CHANNEL: Engineering School Mathematics
© 2015 Scott Baldridge and Ben McCarty

Meet Mathematician Jeremy Van Horn-Morris

Intended Audience: College students, and high school students who think they may be potential math geniuses.

In this episode we meet Jeremy Van Horn-Morris, a mathematician from the University of Arkansas, who talks to us about some geometric and visual tools mathematicians use to understand questions in classical physics concerning the motion of particles.

Jeremy discusses some of the motivation behind his paper with Kenneth Baker and John Etnyre, Cabling, contact structures and mapping class monoids.  Here is the abstract to their paper:

In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

While the video above is for a general audience, Jeremy Van Horn-Morris’s paper is not (it’s written for other mathematicians).  However, high school students who think they are potential math geniuses may still enjoy looking at it to see what advanced theorems and proofs look like.

Students and mathematicians alike will also enjoy visiting Kenneth Baker’s blog, Sketches of Topology.  The post, Its full of surfaces, provides a stunning visualization and description of the open book decomposition coming from the trefoil knot, which was mentioned by Jeremy the end of our interview.  Some additional posts containing phenomenal depictions of open book decompositions can be found here and here.