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:

Snap

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
Lead Writer and Lead Mathematician, EngageNY Mathematics Curriculum
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
http://ramosgroup.squarespace.com (this article can be found at this website)

[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 

Posted in Commentary, Engineering School Mathematics | Tagged , , , , , | Leave a comment

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.

Finally, it should be said unequivocally that the goal is for children to learn their multiplication table from 0x0 to 10×10 so that they no longer need to think about how to derive the answer each time–that they fluently recall each fact from memory without thinking. The process outlined in these videos, if practiced regularly, will lead to that instant recall over time (be patient). But, and this is important, the process also helps  children to think flexibly about numbers in general (while practicing addition/subtraction). For example, Autumn was able to apply the same technique to 18×9 in the video where she had to find 180-18 in her head—just finding 180-18 is a sizable task for a first grader. That flexibility and number sense will help your children as they starting working with letters in Algebra just a few years later, where that flexibility is required, even demanded.

As always, comments are welcomed. Also, please share on twitter/Facebook/etc.

CHANNEL: Growing up with Eureka

Posted in Growing Up With Eureka | Tagged , , , , , , | Leave a comment

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

Video | Posted on by | Tagged , , , , , | 1 Comment

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

Video | Posted on by | Tagged , , , , | 2 Comments

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

Video | Posted on by | Tagged , , , , , , | 3 Comments

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:

http://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:

fundamental group
complex projective plane
line arrangement
intersection lattice
moduli space

Channel: Geometry and Topology Today

Video | Posted on by | Tagged , , , , , , , | Leave a comment

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.

Meet Adam Baker (5th grade teacher), Colleen Sheeron (2nd grade teacher), and Hae Jung Yang (1st grade teacher).

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

Posted in That's News To Me | Tagged , , , , , | Leave a comment

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

Posted in Geometry and Topology Today | Tagged , , , , , | 2 Comments

100th Day of School

Intended Audience: Teachers and parents of K-5 students

On the 100th day of first grade, Autumn asked me if we could do a Growing Up With Eureka based upon the day.  Well, yes, of course!  And a few minutes later we were exploring different ways to skip count to 100. Watch us stumble and recover when we skip count by fours.

CHANNEL: Growing Up With Eureka

Video | Posted on by | Tagged , , , | 1 Comment

Making Science Cool: Solving the Shortage of Math and Science Students

In 2011 I was delighted to be invited by U.S. News and World Report to sit on a panel to discuss how to get students excited about Science, Technology, Engineering and Math (STEM) and how to stimulate their interest in careers in these same disciplines.

It was quite exciting–panel participants and speakers included two governors (John Engler and Gaston Caperton), an astronaut (Anousheh Ansari), a fashion designer (Marc Ecko), a science photographer (Felice Frankel), and a number of others that you can find out more about by going here.

Part I of the days events include talks by Mortimer Zuckerman, Governor Gaston Caperton, and Governor John Engler:

Part II is the panel discussion that I participated on with a number of outstanding people:

The other people in this video include: Brian Kelly of U.S. News and World Report, Anousheh Ansari of Prodea Systems, Marc Ecko of Marc Ecko Enterprises, Felice C Frankel of Massachusetts Institute of Technology, Tom Luce of National Math and Science Initiative, Paul Powell of True North Troy Preparatory Charter School, and Linda P Rosen PH.D. of Change the Equation.

I’d like to personally thank Mort Zuckerman and James Long of U.S. News and World Report for inviting me to participate.  It was quite an honor!

You can view a news brief of the days events by watching the video below or by reading Experts: STEM Education Is All About Jobs by Jason Koebler.

CHANNEL:  That’s News to Me

Posted in That's News To Me | Tagged , , , , , , , , , , , , | Leave a comment