## 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.

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

## Skip Counting with Fractions

Intended Audience:  Grades 3-6 math teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn skip counts by fractions 1/2, 1/3, and 1/5.  There are a number of ways to skip count by a fraction.  Here are some of them used in Eureka Math/EngageNY:

1. 1/4   2/4   3/4   4/4   5/4   6/4   7/4   8/4   9/4 …
2. 1/4   2/4   3/4   1   1 1/4   1 2/4   1 3/4   2   2 1/4 …
3. 1/4   1/2   3/4   1   1 1/4   1 1/2   1 3/4   2   2 1/4 …
4. 1/4   1/2   3/4   1   5/4   3/2   7/4   2   9/4 …

Autumn is doing the second skip counting technique above (the fourth is the hardest, which is why it shows up in later grades—try it with 1/6).  The beauty of the second skip counting technique is that

• it emphasizes the whole unit “…, 3 fourths, ONE, ONE and 1 fourth, ONE and 2 fourths, …”
• it emphasizes the repeating pattern of important fractional units (1/4, 2/4, 3/4)  between each whole unit.

That doesn’t mean the other skip counting techniques are not important! They all have a role to play in a curriculum.  For example, the first skip counting technique emphasizes that counting fractions is just like counting whole numbers but in a different unit.  That is, “1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths,…” is just like “1 apple, 2 apples, 3 apples, 4 apples, 5 apples…”

Obviously, Autumn already knows a lot about fractions.  I apologize for not showing how to develop the concept of a fraction (maybe another video?).  This process takes a long time and is carefully developed in the Eureka Math/EngageNY curriculum.  You can find out more about how we do this in the curriculum by reading Chapter 6 of “Elementary Mathematics for Teachers” that I co-authored with Thomas Parker.

Regardless, there are many things that parents can (and often already do!) with their children to help them get ready for fractions.  These things include very sensible activities like using a tape measure or cooking cups where the notion of fraction just naturally manifests itself, “Honey, measure out 1/3 cup of sugar please.”  Early on, “1/3” is basically only an adjective modifying the noun “cup;” it references a particular measuring cup, but even so it does bring up a nice way to have a discussion about meaning of those fractions with your children.  Tape measures are also great, “What are those marks between 1 inch and 2 inches on the ruler? Between 5 inches and 6 inches? What could they mean?”

Surprisingly to me, one of the main paths that Autumn came to understand fractions was from reading to her.  Here’s the story:  I started reading full-length novels to her starting when she was 2 years old (stories like Narnia, Lord of the Rings, Harry Potter, Watership Down, etc.).  These are thick books that are close-to or over 1000 pages each.  As we read each book, I started (rather by accident) to discuss with her the fraction of the book that we had read, “Look, honey, we are 2 thirds of the way through!”  An unanticipated-but-nice feature of thick books is that it is very easy to split the book’s pages into thirds, fourths, fifths, sixths by separating the pages with your fingers.  Since the books were all of different thicknesses, over time Autumn came to see the main issues in defining fractions:  to establish the whole unit and the relationship of the fractional unit to that whole unit (cf. how fractions are developed in grade 3 of the CCSS).

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

## Ben McCarty plays “Swing that Hammer”

Mathematicians have many talents! In this episode, mathematician Ben McCarty plays and sings the song, “Swing that Hammer.” Ben is a professor of mathematics at the University of Memphis, and the lead mathematician for grades PK-5 of the Eureka Math/EngageNY curriculum.  He coauthored the article “Fluency without Equivocation.”

Ben is here at LSU this week working on a new theorem with me on “special Lagrangian cones,” a type of object that is helpful in studying mirror symmetry from theoretical physics.  I was able to cajole him into playing a song while I recorded.  Ben’s favorite instrument is the banjo, which is probably why his email address is “banjoben.”  In this video, Ben is playing a Breedlove Pro Series C25/CRH guitar.  Enjoy!

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

## Remarks on the History of Ratios

The idea that a ratio is a pair of magnitudes is in Euclid (fl. 300 BC), Elements, Book V.  It is interesting to note that Euclid says that a ratio is the relationship between two magnitudes, not the pair itself.

Greek mathematics did not have the explicit concept of equivalence relation, but the “conceptual grammar” of Greek math is most easily understood by us if we describe it using the idea that they recognized certain equivalence relations “implicitly.”  Consider Euclid’s definition of angle:

“the inclination to one another of two lines in a plane which meet one another and do not lie in a straight line”

Clearly, Euclid means something that can be the same in two different pairs of lines.  This pair of lines and that pair of lines may have the same inclination, and if so, they are equivalent. The same thinking applies to ratio: two different pairs of magnitudes may stand in the same ratio. The genius of ancient Greek mathematics is to produce an operational definition for sameness. Book V, Definition 5 states:

“Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.”

One thing that I find interesting about the discussion on the notations of ratios—and it’s an issue that occurs often in the analysis of school math—is the contrast between the thing itself, and the means by which we make reference to it. The most basic example is the distinction between the written symbol and the thing the symbol refers to: the number 6 (which, according to Plato, the soul observes before birth—but you can take its ontological status to be what you prefer) and the numeral 6, which is a mark on paper.

As for ratio, is it a number? A pair of numbers—or pair of quantities—by means of which we refer to a number? Or an equivalence class of pairs of numbers? One can find advocates for each of these conceptual images.  The Progressions Document on Ratios and Proportional Relationships has chosen to view a ratio as an ordered pair of real numbers, not an equivalence class of such pairs, nor any of the other options.

Because of history, the word “ratio” has many meanings. I haven’t even mentioned the “Rule of Three” and the way that has shaped the way we talk about proportion. We can acknowledge that there have been many different traditions.  But if we don’t agree on a simple, direct language, we will wind up like the poor servant in the comical fairy tale of the “Master of All Masters.”  What we need is a firm grasp of an idea and the ability to provide an account of how we are using our words.  There is no need to preserve odd notions from Textbook School Mathematics.

For the purposes of presenting math clearly, we must attach meanings to words in only one way.  We can control our own classrooms and the language used there, but we cannot do anything about the fact that in the world there is a mixture of habits, traditions and perspectives, and everyone will surely encounter that at some point in some way or another.  After students have mastered one way of talking about things, they may find it convenient or even necessary to consider or use other ways.

CHANNEL: Engineering School Mathematics

## Subtraction Problems with Kittens

Intended Audience:  Grades K-5 teachers, prospective teachers, and public/private and homeschool parents.

In this video, 6-year-old Autumn shows different methods for subtracting in the context of the word problem, “If there are XXXX kittens in a barn and YYYY are adopted, how many are left?” Watch as the question degrades quickly!

The first question, answered by finding 17-8, is done using the number bond “8 is 1 and 7.” First, take away 7 from 17 to get 10, then take 1 more to get 9.

This is one of the “bread-and-butter” methods of Eureka Math because it also helps teach place value (subtract to 10, then subtract the rest).  To prepare students to use this method (including Autumn!), a lot of work done in PK-1 centers around 10 frames:

This one picture shows many number relationships all at once.  It corresponds to the “hand number line” in the “Learning to Multiply, Part I” video (e.g., the top row corresponds to the left hand).  It shows the number bond “9 is 5 and 4” (1 left hand and 4 right-hand digits).  Most important for the subtract method that Autumn used, it shows the number bond “10 is 9 and 1” (note the empty space can be counted too!).  Autumn has done enough work with 10 frames that this picture is one of the pictorial representations she can visualize when doing subtraction calculations.

The answer to the second problem, 53-18, is solved using a different method. In this problem, Autumn sees that 18 is close to 20, and that 20 is easy to take away from 53: 53-20=33. She took 2 too many, though, so adds those 2 back in to get 35. As an exercise, try to draw this on a number line yourself.

The third and final method shows up in answering 114-96.  Autumn imagines 96 and 114 on a number line. She then knows that the difference is just the distance between the two numbers, which is easily found by backing up 14 to go from 114 to 100, and then another 4 from 100 down to 96:

The total distance is 18, which means:

114-96=18.

The final question is just the third method used again, and in this case, it is even easier to see: 1017-999 = 17+1 = 18.

Finally, let’s talk about the question, “If there are 1017 kittens in a barn, and 999 are adopted, how many are left?”  In the Eureka Math curriculum, this is what I started calling (and which the writers have come to affectionately use as part of their vernacular):

Completely Ridiculous Artificial Problems

If used in isolation, the 1017-999 word problem in the video is absolute C-R-A-P.  It’s so ridiculous that every student would see it as artificial.  The writers of Eureka worked very hard to not inadvertently write C-R-A-P because it sends the very negative message that “math is only useful for ridiculous, artificial problems.”  Sadly, one of the reasons many students get turned off to math is due to all the C-R-A-P in the standard Textbook School Mathematics (TSM) curricula in the U.S.  If a large enough percentage of math problems are C-R-A-P, students are likely to judge that the entire enterprise of mathematics is ridiculous and artificial as well.

But, as this video shows, one can delve into the world of C-R-A-P if the teacher is honest with their students that the problem is ridiculous and made-up.  In the video, we build up to the C-R-A-P problem by starting with a reasonable question (barns often have 10-100 cats due to so many mice eating grain), and slowly making the problem worse.  The C-R-A-P problem then helps students understand what is a reasonable math question and what isn’t (while having a bit of fun at the same time).  Enjoy Autumn’s expressions as the problems get more ridiculous.

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

## Ratios, ordered pairs versus points, proportional relationships, and proportions

My NCTM talk last Friday (April 17, 2015) generated quite a bit of social media discussion. I had a Twitter discussion with Bowen Kerins and Bill McCallum that was very interesting, but I thought there were a few (twitter-induced?) misunderstandings that I’d like to clear up.

## Ratio Definition and Notation

What is the context in which the notation (2,5) is used for describing a ratio?

As I said in my talk, we must go back to definitions.  The definition of ratio I used for the Eureka Math/EngageNY was based directly upon the progressions document “6-7, Ratios and Proportional Relationships.”  The progressions documents are a set of companion documents to the Common Core Math Standards.  While the progressions documents are not the actual standards (and are in complete-but-still-draft form), they provide guidance in creating curricula that meet the Common Core Math Standards.  Here is a picture of the definition of ratio from that document (page 13):

This definition needs a bit of translation to write it without the notation A:B.  “Pair” in this definition means “ordered pair,” which is given by the order of A and B in the notation A:B.  A literal restatement of the definition of ratio above without notation is:

ratio is an ordered pair of non-negative numbers, which are not both zero.

Note: Neither the progressions document’s definition nor the restated definition mentions  equivalence classes of ordered pairs of numbers.  That is, 2:5 is a different ratio from 4:10. This distinction between 2:5 and 4:10 is important and useful pedagogically (for example, it makes it easy and natural to refer to “a set of equivalent ratios” as a grouping of many different-but-equivalent ordered pairs, as is done over-and-over in the progressions document).

Now let’s talk about ways to notate ordered pairs.  When talking about ratios, it is common to notate an ordered pair of numbers 2 and 5 by 2:5.  But here is another perfectly valid way to notate the same ordered pair: (2,5).  In fact, the notation (2,5) is the most commonly accepted mathematical way to notate an ordered pair of numbers (cf. here for equivalent definitions and notation of ordered pair).  In a middle school curriculum we actually want both notations and other notations as well (for example, a column/row in a ratio table) to describe a ratio, depending upon context of course.  I promise to explain why below but let’s look at the confusion first.

## Point versus Ordered Pair Confusion

I think the possible confusion generated on twitter and my talk may have occurred because people were substituting “point” in their mind for “ordered pair.”  The ordered pair (2,5) corresponds to a point in a coordinate plane—but, it is only a correspondence: Ordered pairs are generically different than geometric points.  Mathematically, an ordered pair is a general term for a set of two objects in a given order (again, see definitions here).  For example, the notation (M,N) where M and N are two 3×3 matrices is also an example of an ordered pair in mathematics.  Thus, an ordered pair does not automatically mean it is a point in a plane!  In the presence of a coordinate plane, however, it is safe to blur (and we often do) the distinction and refer to the ordered pair of two numbers as a point.

Here is where I must apologize to Bowen and other attendees of my talk:  I was very, very careful about this distinction throughout the talk but I did not make that distinction explicit.  I referred to the ordered pair (2,5) as an ordered pair.  I did not say that ratios (as ordered pairs) were geometric points until we got to the slides that showed the graph of a proportional relationship.  The graph puts us in the context of a coordinate plane where it becomes safe to blur the distinction between an ordered pair and a point.

Thus, Bill McCallum is absolutely correct when he said in a tweet:

“A ratio is an ordered pair in a certain context; I wouldn’t say [the point] (2,5) is a ratio without context.”  (The phrase “the point” was part of another tweet that Bill was commenting on.)

I too wouldn’t say the “the point (2,5)” is a ratio without context.  Of course, Bill McCallum would probably also say, and I would agree, that one only really uses the notation (2,5) for ratios in the context of proportional relationships, which we will talk about next.

## Proportional Relationship Definition

We are getting closer to the moment where we can explain why having multiple notations for ratio is so very useful.  But first we need to clear up another possible confusion about what a proportional relationship is according to the draft progressions document.  One of the questions asked on Twitter was,

“Is a proportional relationship a set of equivalent ratios? … I’m confused.”

Here’s a picture of the definition in the progressions document (page 14):

The two definitions are synonymous: Set is another word for collection, ratios are (ordered) pairs of numbers, and two ratios are in this set if they are equivalent.  Mathematically, we are just using synonyms to say the same thing.  You can read more about proportional relationships here.

## Why it is useful to have multiple notations for ratio

With the definition of proportional relationship understood, we are finally ready to see the huge benefit of having different but equally valid ways to notate ratios.  Sometimes it is useful to write a ratio as 2:5, like when we write  a single ratio in a word problem.  But when writing down a proportional relationship, it is useful to write a set of equivalent ratios as

{(2,5), (4,10), (6,15), (8,20), …},

and because of that notation, it is even easier to see what to do with this set of ratios when graphing it in a coordinate plane.  In grade 6 and 7 of the Eureka Math curriculum, proportional relationships like {(2,5), (4,10), (6,15), (8,20), …} are initially written as ratio tables.  But there is an important teaching sequence that goes from ratio tables to ordered pairs to plots of points of a graph of a proportional relationship, and the use of the (2,5) notation helps facilitate this transition without getting bogged down in ugly pedantic semantics about notation.

## Proportions

While we are at it, let’s clear up one more thing that came up as a question during and after the talk: the term “proportion” and the difference between “equal” and “equivalence.”  What is a proportion?  For two ratios with well-defined values, a proportion is a statement of equality between the values of the ratios (i.e., an equation).  If you do a search of the progressions document you will see that this is exactly how the term proportion is used in each and every case.  Why use the values?  Because of the difference between when two ratios are equal and when they are equivalent:

• For numbers a,b,c,d, the statement a:b=c:d is true if and only a=c and b=d are true.  Example:  2:5=2:(4+1), but 2:5≠4:10.
• For numbers a,b,c,d, the ratios a:b and c:d are equivalent if there is a number r such that a=rc and b=rd.  Example:  2:5 is equivalent to 4:10, and 2/5 = 4/10.

By using values we get around the need for having two different meanings for the equal sign with regards to ratios (see my post here about how important it is to use the equal sign consistently).  For the brave-of-heart:  Mathematicians have special notation to get around this problem with special notation for the “class of equivalent ratios,” see the use of [2:5] in the introduction to Projective Space.

## Conclusion

Overall, it’s my opinion that the progressions document writers got the conceptual image of ratio essentially correct (for many pedagogical reasons not listed in this post, actually), but they could have been a little bit more clear about how they were using the word “pair” in the progressions Document.  Hopefully this will be cleaned up in the final version of the progressions document (which is still in draft form)–maybe by removing the notation from the definition of ratio (to make the definition notation independent) and using the term “ordered pair” instead of just pair.

As I said in the talk, I certainly empathize with teachers who have thrown up their hands at some point and said, “6.RP.A.1 doesn’t make any sense.”  But the main point of my talk was that if you understand the conceptual images and definitions that the CCSS and progressions writers were using, then it does make sense!

CHANNEL: Engineering School Mathematics

## A big NCTM thanks and next year’s talk

Thanks to everyone who came to my talk at NCTM, especially on a Friday afternoon during happy hour: You are some hardcore rule-breakers! (See question#4 here)  It was a joy to make so many new friends.  I hope you liked the talk and got something out of it. In fact, look for my next post soon that will do a deeper analysis of the terms ratio, ordered pair, and proportional relationship.

## Next Year’s Talk:

My 6-year-old daughter’s pleasurable learning antics has inspired me to consider a talk where we “do math” together on stage at next year’s NCTM meeting, and show off some of the techniques used in Eureka Math.  You can see some of her pleasurable learning in the videos below (she is my co-teacher in this series).  Let me know if you would like to see Autumn in the comment section at the end of this post, or feel free to suggest a topic for me to speak about.  The deadline is coming up quick, so let me know soon!

CHANNEL: That’s News to Me

Posted in That's News To Me | 5 Comments

## I’m at the NCTM meeting this week and would like to meet you!

Want to talk with the lead writer and mathematician of the Eureka Math/EngageNY curriculum?  Here’s your chance to do so at the NCTM national meeting.

I’m scheduled to be at the Eureka Math booth #1308 & #1309 at the following times:

• Thursday: 11:00–2:00 pm
• Friday: 9:30–11:00 am, and 2:00–3:00 pm.  UPDATE: I can no longer meet at 2:00pm.  I may be there later, but I should be in 104C a little after 3pm.

You can’t miss the Eureka Math booth—it’s the one with the classroom-like feel and the cool video graphics on the wall.  Definitely come by and share with me your stories about students learning.

IMPORTANT:  Don’t miss my talk on Friday from 3:30-4:30pm in Room 104C (BEC) on the

Mathematical Secrets behind the Common Core State Standards

Abstract:  Have you ever read a CCSS standard and wondered, “What was the thought behind that standard?” Hear the mathematical meanings behind some of the ratio, rate, and function standards, why they are important, and how those meanings can lead to effective teaching innovations that will help your students to see math as a coherent whole that makes sense.

Presentation Format: General Interest/All Audiences Session
Grade Band Audience: General Interest/All Audiences

FAQ about my talk on Friday:

(1)  I’m an elementary teacher.  Should I attend your talk Scott?

Answer: Absolutely!  In this talk I will describe how vitally important your work is in A Story of Units (grades PK-5) for helping middle school students understand ratios and rates.

(2) I’m a high school teacher.  What’s in it for me?

Answer: Well, converting quantities into measurements, and measurements into numbers is a major step towards studying real-valued functions with real number domains, which is the main theme of A Story of Functions (grades 9-12).  Read my article here for more info.  Plus, rates are the first step towards differential calculus—yes, it’s that important (we won’t be talking about calculus though).

(3) I’m a middle school teacher.  Help!  What exactly is a proportional relationship?  A unit rate?

Answer: These questions are at the heart of the math content of my talk.  The talk will help you look at middle school and A Story of Ratios (grades 6-8) in a whole new way.

(4) Is this talk going to be boring?

Answer: I have a simple test that you can take to determine whether or not you will find my talk boring. To take the test, just follow this one, simple instruction: Stop reading this paragraph right now–not another word.  Couldn’t stop could you?  You are still reading this paragraph, aren’t you?  I fully have your attention now and you couldn’t stop even if I asked you to again, which I won’t.  And here’s the great news–we just got rid of all those mindless, instruction-following, boring people who did stop reading.  The rest of us rule-breakers are now guaranteed to have a good time at my talk!

CHANNEL: That’s News to Me

## Meet Mathematician Aaron Lauda

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 Aaron Lauda, a mathematician from the University of Southern California, who shows us how to represent complicated expressions and equations using pictures. Enjoy! In fact, Aaron has provided more artwork at his website.  Go check it out.

Aaron explains the motivation behind his paper with Mikhail Khovanov, “A diagrammatic approach to categorification of quantum groups I.”  Here is the abstract to their paper:

To each graph without loops and multiple edges we assign a family of rings. Categories of projective modules over these rings categorify U−q(𝔤), where 𝔤 is the Kac-Moody Lie algebra associated with the graph.

While the video above is for a general audience, Aaron Lauda’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.

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636, NSF grant DMS-1249708, and the NSF-funded VIGRE Student Colloquium Committee at LSU.

## ‘Eureka Math’ Embeds Real-World Problems in PreK-12 Mathematics Lessons

Jessica Hughes quotes me in her article:

The article discusses how Eureka Math is a new curriculum for delivering Science, Technology, Engineering, and Mathematics (STEM) education in the United States.

Jessica Hughes asked insightful questions that went straight to the heart of what the curriculum is all about.  You can keep up with Jessica at the Center for Digital Education by following the twitter account: @centerdigitaled.

CHANNEL: That’s News to Me

Intended Audience:  Grades K-2 math teachers and parents.

In this video, 6-year-old Autumn shows different techniques for adding two numbers. Originally the plan was to concentrate on just one technique, but working with a child who already knows multiple techniques means that you just have to “go with the flow.”

The technique we were going to show is based upon the associative property. In the first problem, 7+8, Autumn breaks 8 into 3 and 5 and groups the 7 and 3 together to get 10. The answer is then simple: 10+5 or 15. This method shows off the Associative Property in algebra because we are changing the 3’s association with 5 to an association with 7. This “re-associating” is done symbolically with the parentheses below:

7+8 = 7+(3+5) = (7+3)+5 =10+5 = 15.

For teachers, parents, and students using the Eureka Math curriculum, your students practice this technique through the use of number bonds (the bond has 8 in the “whole” circle, and 3 and 5 in the two “part” circles).

Yes, it’s first grade yet Eureka Math is already preparing your children to be successful in algebra in middle school! Of course, we are not burdening students with words like “associative property” at this stage in their learning.

This is the same technique used in the last problem: 999+64. Autumn takes 1 from 64 and associates the 1 with 999 to get 1000. The answer is then easy:

999+64 = 999+(1+63) = (999+1)+63 = 1000+63 = 1063.

The second technique shows up in the second problem: 6+7. In doing this problem, Autumn says that, since 6+8=14, then 6+7=13 . She knows that 6+7 must be one less than 6+8. Enjoy the look of surprise on my face—I was definitely not ready for that response.

The third technique is easy and was used for 27+12 and 232+232. Autumn realized that, since there was no regrouping, she could add “2 tens + 1 ten = 3 tens” and “7 ones + 2 ones = 9 ones” to get 3 tens 9 ones, or 39. (One does not have to start in the farthest right ones place as in column addition.) I had to help her with the meaning of the digits, like when she said, “2+1=3” and I encouraged her to say “2 tens + 1 ten = 3 tens.”

Mathematically, this technique uses the “Any-order property,” which just means we can arrange addends in a sum in any order with any grouping we want. (It is just repeated applications of the Commutative Property and Associative Property). Symbolically,

27+12 = (20+7)+(10+2)=(20+10)+(7+2)=30+9=39,

The final technique is a combination of the previous techniques when Autumn finds 57+58:

57+58=(50+7)+(50+8)
=(50+50)+(7+8)
=100+(7+(3+5))
=100+((7+3)+5)
=100+(10+5)
=100+15
=115

Autumn, however, doesn’t do this problem exactly like that. It’s hard to tell from the video, but the method she used to find 7+8 was to break each number into 5 plus a number and use the fact that 5+5=10:

7+8=(5+2)+(5+3)=(5+5)+(2+3) = 10+5 = 15.

You get a hint that this is her method when she says “2+3=5.”

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

## Common Core, Eureka Math shake up Louisiana classrooms

Jessica Williams quotes me in her article, “Common Core, Eureka Math shake up Louisiana classrooms” from March 13, 2015.   The article talks about the difference between the Common Core State Standards and curricula that satisfy the Common Core State Standards.

Here’s a quote from her article:

Still, Common Core and Eureka aren’t identical, Munson said. Standards are guidelines for what children should learn and know; curriculums are a way to get there. “Do they meet the standards? Yes,” Munson said. “But they are far from the same thing, and they’ve never been the same thing.”

Jessica Williams has a “nothing but the facts” reporting style that I like and appreciate very much.  She can be followed on twitter at @jwilliamsNOLA.

CHANNEL: That’s News to Me

## Interview with Thomas Lam on electrical networks

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 Thomas Lam, professor of mathematics at the University of Michigan, who studies electrical networks (among other topics).  Thomas gives a great introduction to one of the main problem in electrical networks as well as an application of electrical networks to medicine.  At the end, I ask Thomas when he knew he wanted to become a mathematician.   Did you know that there is a “math olympics” for high school students?

The main problem presented in this video is the motivation behind several papers about electrical networks, including the paper, “Electrical networks and Lie theory,” by Thomas Lam and Pavlo Pylyavskyy.  Here is the abstract to their paper:

We introduce a new class of “electrical” Lie groups. These Lie groups, or more precisely their nonnegative parts, act on the space of planar electrical networks via combinatorial operations previously studied by Curtis-Ingerman-Morrow. The corresponding electrical Lie algebras are obtained by deforming the Serre relations of a semisimple Lie algebra in a way suggested by the star-triangle transformation of electrical networks. Rather surprisingly, we show that the type A electrical Lie group is isomorphic to the symplectic group. The nonnegative part $(EL_{2n})_{\geq 0}$ of the electrical Lie group is a rather precise analogue of the totally nonnegative subsemigroup $(U_{n})_{\geq 0}$ of the unipotent subgroup of $SL_{n}$. We establish decomposition and parametrization results for $(EL_{2n})_{\geq 0}$, paralleling Lusztig’s work in total nonnegativity, and work of Curtis-Ingerman-Morrow and de Verdi\`{e}re-Gitler-Vertigan for networks. Finally, we suggest a generalization of electrical Lie algebras to all Dynkin types.

While the video above is for a general audience, Thomas Lam’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.

CHANNEL: Geometry and Topology Today
© 2015 Scott Baldridge and David Shea Vela-Vick
Supported by NSF CAREER grant DMS-0748636, NSF grant DMS-1249708, and the NSF-funded VIGRE Student Colloquium Committee at LSU.

## EdReports.org’s Review of Eureka Math

Here’s the quote from Education Week that summarizes the report in a nutshell:

In all, just one curriculum series stood out from the pack. Eureka Math, published by Great Minds, a small Washington-based nonprofit organization, was found to be aligned to the Common Core State Standards at all grade levels reviewed.  edweek.org

The full report can be found at edreports.org.  A graphic showing the overall alignment for all curricula can be found here.

As the lead writer and lead mathematician of the Eureka Math curriculum, I just want to take this opportunity to thank all of the wonderful writers who worked so hard on this project.  Let’s keep listening to the teachers and parents who are using our curriculum, and use their advice to make it even better.

I also agree with Lynne Munson, President and Executive Director of Great Minds, when she says:

The teachers who wrote Eureka Math have so much to be proud of today.  Indeed, Eureka is exemplary because the people who wrote it are extraordinary.  Eureka is the result of a historic collaboration between teachers and mathematicians, who know the standards, the math, and the best practices for teaching students.  Every student deserves a great math education, because every child is a great mind in the making. —Lynne Munson

CHANNEL:  That’s News to Me

## The Mardi Gras Problem

Intended Audience: Grades 4 – 8 math teachers and/or home school parents, or anybody who is interested in Mardi Gras parades!

We enjoy going to family-friendly Mardi Gras parades in Louisiana. These parades are exciting events filled with floats where riders throw beads and stuffed animals to children.  At the 2015 “Krewe of Artemis” parade in Baton Rouge, Autumn and I wondered:

## Fluency without Equivocation

Intended Audience: Parents, teachers, and other educators involved in the EngageNY Mathematics Curriculum. Download a PDF of “Fluency without Equivocation.”

by Scott Baldridge, Ben McCarty, and Robin Ramos

For many years a passionate group of math educators has decried the memorization of math facts in grades K-5 as unproductive for learning mathematics.   Indeed, the larger education establishment has long known that “blind memorization,” i.e., handing students a list of random facts and drilling them with a timed test until the facts are memorized, is no substitute for helping students to memorize their facts through activities that develop their number sense.  There can also be negative consequences to giving such “drill and kill” tests.  For example, these types of timed tests often give students the wrong impression that, in mathematics, “speed” means “smart.”  The point: the methods employed to help students memorize and fluently use math facts matter a great deal to their overall understanding and creative use of mathematics in their lives.

Recently, some educators in that passionate group have taken one sentence out of 20,000+ pages of the EngageNY math curriculum and inferred from that sentence that the entire curriculum approaches fluency only through “blind memorization.”  As the lead writers and mathematicians of the EngageNY curriculum, we feel that the curriculum has been unfairly characterized—that we have been accused of perpetuating the very thing we carefully designed the curriculum to avoid.  Therefore we have written this article to reaffirm, without equivocation, the following two points for parents, teachers, and other educators involved with the EngageNY curriculum:

• An important goal of the EngageNY curriculum in grades K-5 is for all students to become fluent with the math facts (addition tables, multiplication tables, algorithms, etc.).  By fluent we mean students can recall facts without hesitation and can perform routine calculations without thought—similar to speaking a language fluently.
• We intentionally engineered the curriculum to reach this goal through joyful-yet-rigorous activities that develop students’ number sense, not through “drill and kill” blind memorization.  When it comes to the importance of number sense, we are in complete agreement with the educational establishment as a whole.

In the first part of the article we explain what it means to develop number sense, and why it is important in reaching the goal of helping students become fluent with their facts.  The second part of the article shows three examples of activities that we use to develop number sense.

## Developing Number Sense

The sentence that the educators quoted can be found in documents describing the instructional shifts, including the fluency section of the How To Implement document for A Story of Units, which is a document about the PK-5 portion of the EngageNY curriculum.  The fluency component of each lesson is further explained in the How To Implement document as having the following purpose:

“Fluency is designed to promote automaticity by engaging students in practice in ways that get their adrenaline flowing. Automaticity is critical so that students avoid using up too many of their attention resources with lower-level skills when they are addressing higher-level problems. The automaticity prepares students with the computational foundation to enable deep understanding in flexible ways.” (page 22)

Flexibility and automaticity are key here.  A skilled musician, surgeon, athlete, or chef has certain core processes so practiced that they become automatic, thereby freeing up the brain to focus on the larger task, and adapt to the moment.  We want students to be able to do the same with mathematics—to take their knowledge of mathematics and use it.  For example, a third grader who already knows that $5\times 4$ is $20$ is then able to reason that $7\times 4$ is just 2 more fours, and therefore  can be broken down into two “easier” parts that they already know from memory:

$7\times 4 = 5\times 4 + 2\times 4$

This is the distributive property in action, which becomes an important concept later in algebra.  The student is making use of structure, making use of known facts that can be recalled from memory.  Eventually, $7\times 4$ becomes a known fact too, so that when encountering a problem like $57\times 34$  in a later grade, students are not stumbling over how to multiply $7\times 4$  but rather, simply recalling it from memory, able to focus on the process of multiplying two 2-digit numbers.

Indeed, one of the articles the educators cite in their criticism of EngageNY actually supports this sensible approach to fluency we took in writing the curriculum.   In the research article titled, “Learning by strategies and learning by drill—evidence from an fMRI study,” some subjects were trained by blind memorization, while others were trained with a variety of back-up strategies.  Both groups were instructed, “to work as fast and accurate as possible.” Having tested both groups, the researchers concluded that:

“Though there is no doubt that skilled and automatic retrieval of arithmetic facts is advantageous in calculation tasks, saving working memory resources, time, and effort, the way to reach this goal should start with back-up strategies providing the understanding of the underlying numerical relations.”  (Delazer, Ischebeck, Domahs, et al, NeuroImage, 2005)

We designed activities in the curriculum that develop skill, flexibility, and automaticity. Good fluency activities engage students in flexible thinking and help them develop their number sense, while pushing them toward the ability to recall key facts from memory.  Throwing out the good fluency activities along with the bad and expecting the student to learn math anyway would be like expecting someone to play baseball without developing the ability throw and catch a ball, pick the banjo without developing the skills needed to pick, or conduct surgery before learning to use a scalpel. Certainly bad fluency activities should be eradicated, but not at the expense of the good.

## Examples

Because we basically agree with the passionate group about the difference between bad and good fluency, the negative criticism levied against the EngageNY came as a bit of shock to us.  If the educators who made the accusation about the EngageNY curriculum had actually looked at the curriculum materials, they would have seen for themselves that it is brimming over with mental math, counting, and arithmetic activities that develop mathematics with understanding.  In what follows, we’ll describe three ubiquitous fluency activities from the Engage NY curriculum that exemplify the development of automatic retrieval through the process of understanding underlying numerical relationships:

### The Sprint [1]

At first glance, the Sprint looks quite similar to the timed test many rightly criticize.  Both are timed, but the structure and intentional design of a Sprint makes it a completely different experience from “drill and kill” tests.

When administering a Sprint the teacher distributes the first of 2 analogous problem sets (called Sprint A and Sprint B).  The students are given 60 seconds to complete as many problems as they can of Sprint A.  Next comes a short-but-focused period of time where students analyze the problem set: the Sprint is intentionally structured to encourage students to look for patterns in the problems—patterns that will reappear in Sprint B.  For example the following sequence of problems comes from a Sprint in Grade 3, Module 1:

1. 5 + 5 = ___
2. 2 fives = ___
3. 2 + 2 = ___
4. 2 twos = ___

The patterns and relationships from one problem to the next are investigated, articulated by the class as a whole, and used by each student to their advantage in completing Sprint B.  Of course, the patterns that students discover are the very number relationships that help build their number sense.  Finally, students take and correct Sprint B, and report how much better they did on Sprint B than on Sprint A.

Sprints intentionally move from simple to complex problems so that the lowest performing student can always have success with the earlier problems, and the highest performing student is unable to complete all the problems. The goal is not for students to complete a set number of problems in a set time (it’s not even given a grade), but rather for students to become self-aware of their own improvement.

Thus, students come to learn that they are competing with themselves, which focuses the student on a growth mindset.  This alleviates the “speed” anxiety that students often experience with “drill and kill” fluency exercises, but still allows the ticking clock to generate excitement and adrenaline while providing a real way for students to see their own personal growth.

Is memorization one of the long-term goals of Sprints?  Absolutely!  The Sprints are dealing with ideas that the students will need to use as stepping-stones for understanding later on.  But the approach is not blind memorization, but rather the achievement of automaticity through understanding numerical relationships.  Having already been introduced to the conceptual underpinnings of the math content by the time the Sprint is given, the Sprint provides an opportunity to practice to automaticity the numerical relationships needed to build the student’s number sense.

### Snap

One of the fluency games that the passionate group of educators suggests is called Snap It, where students take a linker cube train with a specified number of cubes.  On the signal, they break the train into two parts, hide one part behind their back and then the other children have to work out what the missing part is.  We whole-heartedly agree with this activity and include something very much like it in the curriculum.  The following application problem comes from Module 4 of Kindergarten:

The game is introduced in an application problem, but is also continued later via fluency activities.  Of course, the key skills developed by this game, namely the ability to flexibly decompose numbers, find the missing part, or total, are practiced in numerous other fluency activities as well.

### Skip-Counting

This activity shows up throughout A Story of Units in many different contexts.  Students early in elementary school learn to count, then to skip-count by 10, 5 or even 2.  Later they learn to skip count by 3.  Later still they learn to skip count by unit fractions, or even measurement units, and include simple conversions.  For example students might chorally count together:  “1 fourth, 2 fourths, 3 fourths, ONE, 5 fourths, 6 fourths, 7 fourths, TWO!”

Such activities emphasize that even as the unit being counted changes, e.g. counters in Kindergarten, tens in Grade 1, twos and hundreds in Grade 2, fours and fourths in Grade 3, ten thousands in Grade 4, and volume units in Grade 5, we still work with all of these quantities in the exact same ways.   These skip-counting activities are also used to help students utilize important mathematics, e.g. the relationship of skip counting to multiplication (“When I skip counted by four to find 3 fours, I got to 12. Three times 4 is 12!”).  It also helps students learn the distributive property (“I see that 6 sevens is 42, and 7 sevens is just one more seven, so 7×7=49”).[2]

These are just a few of the many types of fluency activities one can find in the EngageNY curriculum.  Throughout, all fluency activities are designed to help students see relationships, and thereby develop number sense.  Their increased number sense subsequently leads to the development of recall of certain key math facts. Will students recall things at different speeds?  Of course.  But recall is certainly simpler than always having to rely upon some multistep strategy, and thus will require a lower cognitive load in later grades where it is assumed that students have automaticity.

Let’s eliminate blind memorization as a “teaching technique” but let’s not eliminate automaticity as one of the goals that good fluency activities can achieve.   That’s what we aimed for in writing this curriculum:  to give teachers sensible activities to do in their classrooms that encourage their students to learn math facts to automaticity, and to add, subtract, multiply, and divide fluently.

Scott Baldridge
Associate 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/

Robin Ramos
Lead Writer, PK-5, EngageNY Mathematics Curriculum

[1] Read more about Dr. Yoram Sagher’s Sprints by going here.  Bill Davidson, the author of many of the sprints used in A Story of Units has a nice introduction to Sprints here.

[2] For a demonstration of this idea check out the following video on Growing up with Eureka here.

CHANNEL: Engineering School Mathematics

## Multiplying by 9

Intended Audience: Teachers and Parents of K-5 students.

In this video, 6-year-old Autumn shows how easy it is to multiply by 9. Watch her multiply 18×9 in her head and explain how she did it!

Parents and teachers may also want to watch Autumn’s and my 3-part video series on learning how to multiply along with this video (Part I, Part II, Part III).  In the 3-part series, Autumn shows the basics of learning to skip count while keeping track of the number of skip counts on her fingers. This method helps young children learn what multiplication means and gives them a way to confidently find products of two numbers where one of the numbers is 2, 3, 4, 5, and 10. That, together with the commutative property (i.e., 6×7 is the same as 7×6), leaves the following products:

6×6, 6×7, 6×8, 6×9, 7×7, 7×8, 7×9, 8×8, 8×9, 9×9.

This list can be reduced to just 6 facts by learning how to multiply by 9, i.e., the content of this video.  The multiplication by 9 method in this video can be easily seen using unit math: 9×7 means finding  “9 sevens.”  But just as “9 apples = 10 apples – 1 apple,” the same holds for sevens:

9 sevens = 10 sevens – 1 seven.

Of course, 10 sevens = 70 is easy, so 9×7 = 70 – 7.

As you watch Autumn, note that an important prerequisite to this technique is how to take away a 1-digit number from a multiple of 10, for example, 70-7, 80-8, 90-9, etc. This skill in turn comes out of learning to work with “10 combinations,” i.e., 2 and 8 make 10, 3 and 7 make 10, 4 and 6 make 10, etc. All of these prerequisite skills are learned and practiced in the Eureka Math/EngageNY math curriculum in grades K-2 using joyful mental math/counting activities and number bonds (take a look!).

With multiplication by 9 understood, that only leaves the six “most troublesome” facts:

6×6, 6×7, 6×8, 7×7, 7×8, 8×8.

You can watch Autumn explain in another video how to find some of these products just knowing that “6×7=42” by following this link.

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

## Learning to Multiply, Part III

Audience for this post: Teachers and Parents of K-5 students

In the grand finale and Part III of a three part series, we put the two skills learned in Part I and Part II together to show an easy way to start to learn the basics of multiplying numbers by 2, by 3, by 4, by 5, and by 10.

The three videos show the major “lampposts” along the way: together the videos highlight one of the big subplots of A Story of Units that unfold in the Eureka Math/EngageNY Curriculum in grades K-3. Each lamppost is reached through a variety of mental math/counting activities and pictures that are designed into the curriculum in those grades.

Since this video only shows what it looks like when your students/child has reached the lamppost, I also encourage viewers to make and reply to comments about their experiences in the comment section below.

Next up: We show an easy way to multiply by 9!

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

## Learning to Multiply, Part II

Intended Audience: Teachers and parents of K-5 students

This video is Part II of a three part series on how to start the process of learning to multiply with your child/students.  See Part I here.

The three videos only show the major “lampposts” along the way towards learning to multiply with these methods. To reach each lamppost takes lots of joyful counting and visual activities that we embedded into the Eureka Math/EngageNY Curriculum in grades K-2.

The major lamppost we see in this video is skip counting. Again, the Eureka Math curriculum has many activities designed to help your child (or students) learn how to skip count easily in fun ways.

Since this video only shows what it looks like when your students/child has reached the lamppost, I also encourage viewers to make and reply to comments about their experiences in the comment section below.

CHANNEL: Growing Up With Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Supported by NSF CAREER grant DMS-0748636

## Learning to Multiply, Part I

Intended Audience: Teachers and parents of K-5 students

This Growing Up With Eureka video is Part I of a three part series on how to start the process of learning to multiply with your child/students. By “start the process” I mean we show ways to learn multiplication facts by 2, by 10, by 5, by 3 and by 4 using skip counting and unit math. We don’t show in this series how to teach the six most troublesome facts: 6×7, 6×8, 6×9, 7×8, 7×9, 8×9, or how to use the commutative property to cut the number of facts in half (check here for hints on how to cover the troublesome facts).

The three videos only show the major “lampposts” along the way towards learning to multiply using these methods. To reach each lamppost takes lots of joyful counting and visual activities that we embedded into the Eureka Math/EngageNY Curriculum in grades K-2.

The major lamppost we investigate in this video is how to help students immediately recognize the numbers 1, 2, 3, …., 9 on their hands by using their hands to visualize a “number line.”  Again, the Eureka Math curriculum shows activities involving 10-frames, number paths, and counting exercises designed to aid in learning this recognition.

Since this video only covers what it looks like when your students/child has reached the lamppost, I also encourage viewers to make comments and reply to complements about their experiences in the comment section below.

Check back to see Part II and Part III soon!

CHANNEL: Growing Up With Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Supported by NSF CAREER grant DMS-0748636

## Line arrangements from an advanced perspective

Intended Audience: Mathematicians and high school students who think they are potential math geniuses.

We continue the interview with Moshe Cohen on line arrangements, but now at a graduate student level. To see the earlier interview with Moshe geared at a high school level, go to:

https://scottbaldridge.net/2015/02/02/interview-with-moshe-cohen/

In this interview, Moshe explains the theorem he proved in the paper, “Moduli spaces of ten-line arrangements with double and triple points,” by Meirav Amram, Moshe Cohen, Mina Teicher, and Fei Ye. The paper was supported in part by the Minerva Foundation of Germany through the Emmy Noether Institute and the Oswald Veblen Fund of the Institute of Advanced Study in Princeton. Moshe’s travel back to the United States to produce this video was supported by the European Research Council under the European Union’s Seventh Framework Programme, Grant FP7-ICT-318493-STREP.

First year graduate students (and high school students who think they are potential math geniuses) can investigate some of the words talked about during this interview, including:

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

## Meet some of the writers of Eureka Math

What I love about this short video (other than the obvious discussion of Eureka Math as a curriculum system for PK-12) is that it features three of the many talented teacher-writers that worked on the project with me.

Special thanks go out to you three and all the other teacher-writers who I have had the honor of working with to make learning mathematics a joyful experience.

CHANNEL:  That’s News to Me

## Interview with Moshe Cohen

Audience: Everyone, and especially teachers who want to show to their students a mathematician explaining research mathematics

In this episode we meet Moshe Cohen, a mathematician who studies ways to arrange planes in 4-dimensional space. The interview starts with an easier question that can be answered by any student in any grade. Enjoy!

The main problem presented in this video is the motivation behind several papers, including the paper, “Moduli spaces of ten-line arrangements with double and triple points,” by Meirav Amram, Moshe Cohen, Mina Teicher, Fei Ye. The paper was supported in part by the Minerva Foundation of Germany through the Emmy Noether Institute and the Oswald Veblen Fund of the Institute of Advanced Study in Princeton. Moshe’s travel back to the United States to produce this video was supported by the European Research Council under the European Union’s Seventh Framework Programme, Grant FP7-ICT-318493-STREP.

While the paper is not for a general audience (it’s written for other mathematicians), high school students 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